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 THE 
 
 FOUNDATIONS OF MUSIC 
 
 BY 
 
 HENRY J. WATT, D.PHIL. 
 
 Author of The Psychology of Sound 
 
 Lecturer on Psychology in the University of Glasgow and to the 
 Glasgow Provincial Committee for the Training of Teachers. 
 Sometime Lecturer on Psychology in the University of Liverpool. 
 
 CAMBRIDGE 
 
 AT THE UNIVERSITY PRESS 
 1919
 
 THESE WORKS 
 ON SOUND AND ON MUSIC 
 
 I DEDICATE 
 TO MY WIFE AND HER ART
 
 PREFACE 
 
 IN my previous volume The Psychology of Sound I made a minutely 
 critical analysis of the elementary phenomena of sound and their 
 simpler complexities, and I developed what seemed to me to be the 
 only systematically true and promising theory of these phenomena. 
 The work was necessarily addressed to those who are primarily interested 
 in such a study, i.e. to psychologists and to physiologists. But I 
 endeavoured to make the material as interesting to the theoretical 
 musician as was possible under the circumstances. 
 
 Not that the latter has little interest in such fundamental analysis. 
 On the contrary he is profoundly concerned to know how his art springs 
 from its roots in mere sound and to see that the foundations ascribed 
 to it are such as will evidently suffice to bear the whole superstructure 
 of music. But the purely psychological or 'phenomenal' point of view 
 could not but be new and strange to his mind, requiring some time to 
 come into growth and fruition there. Once the essential nature of the 
 position has been grasped, its spontaneous development is certain. 
 
 There is no inherent difficulty in ascribing volume and order to 
 sounds or to tones. The difficulty springs merely from the unfamiliarity 
 of the object in such connexions. At the present day conviction is 
 much more easily secured for descriptions and theories of material 
 objects even although many of their students may never have come 
 into contact with them at all than it is for descriptions and theories 
 of psychical objects, although their students are almost of necessity 
 constantly face to face with them at any desired moment. Every one 
 who takes any interest in music has had unlimited opportunities of 
 turning his observations upon tones and their sequences and combina- 
 tions. But in the great majority of cases he has seldom, if ever, looked 
 studiously at pictures or models of the sensory organ of hearing and in 
 all probability knows nothing of that organ by direct observation of it. 
 But he will nevertheless drink in a description and theory of the material 
 organ with avidity, while he will turn a bored and sceptical ear to a 
 direct analysis and theory of tones, although for both purposes similar 
 methods and explanations may have been used. The mere postulation 
 of a material thing as the bearer of volumes and orders and their 
 coincidences and overlappings seems to bring a special comfort to the 
 mind.
 
 viii PREFACE 
 
 It would be wrong to suggest that this scepticism is quite general 
 in its scope. .The artist is certainly clearly aware that what he usually 
 judges and accepts or rejects is the direct phenomenal impression that 
 is immediately before his mind's gaze. But when it comes to science 
 and to theory, what he has learnt to crave for is in the main a material- 
 istic exposition. The musician has long since accustomed himself to a 
 theoretical diet of beats, partials, and material-mathematical expressions 
 for intervals. Such things seem real and tangible, as it were. But, 
 after all, they seem so in most cases "only because they are more 
 familiar. 
 
 Much of the difficulty is due also to a widespread shallow attitude 
 towards any scientific aesthetics, an attitude unfortunately greatly 
 encouraged amongst musical theorists by Helmholtz's very unsatis- 
 factory distinction between natural law and aesthetic principles. The 
 mere existence and operation of personally subjective forces that affect 
 our artistic judgments seem to convince so many people that no science 
 of these judgments can be achieved. There is no disputing about 
 tastes, they say. And if a body of critical knowledge can be extracted 
 from established works of fine art, such rules are held to be merely 
 the conventions of the ages that created them. The next genius that 
 comes along may blow the whole system to the winds of oblivion. So 
 it seems to those who are struck most of all by the innovations of each 
 master and have not perhaps the patience to follow out the great 
 purpose that is common to them all and that each merely carries on to 
 finer and finer issues. But a great master is by no means an accident. 
 He is one who, taking himself as a man amongst many, has learned 
 how to construct an enduring object that is far more likely to arouse 
 in others the joys of beauty he has felt and anticipated for them than are 
 the works of lesser minds. His medium is the orderly realm of mind that 
 he shares with his fellows. But its laws are not his creation; they are 
 only his discovery. He has learnt to turn them to his will. The scientist 
 who comes after him has, with his help, to formulate them in knowledge. 
 That knowledge is the science of aesthetics. 
 
 In this volume I have sought more or less evenly to serve the purposes 
 of both the psychologist and the musician. In order to make the work 
 complete in itself up to a certain point I have traversed the ground 
 covered in the psychological part of the earlier volume, omitting only 
 those parts that are of little interest to the musician. All critical dis- 
 cussion has been passed over at this stage, so that the earlier part of
 
 PREFACE ix 
 
 this volume is more or less a careful and straightforward ('dogmatic') 
 exposition of the fundamental notions of the psychology of tone. I think 
 the musician should find it useful and helpful, as also will those who 
 wish to have an exposition of the system without the technical dis- 
 cussions and criticisms. Those who are familiar with the previous 
 volume will hardly find anything new before chapter ix (p. 55). Except 
 in so far as they are interested in straightforward and logical exposition, 
 they may begin the work at that point. The previous pages, however, 
 are not in any sense a mere repetition of the earlier volume. They 
 have been written entirely afresh. Only, as is after all inevitable, they 
 are based on the same body of facts and notions as was the analytic 
 psychological work of the earlier volume. I have not gone into any 
 binaural or physiological problems this time. 
 
 The parts beyond chapter ix are addressed both to psychologists 
 and to musicians. In the preface to the previous volume I said that 
 my " theoretical constructions must be carried somewhat farther before 
 they can be held to have passed fully over into the elements consciously 
 used [and known] by productive musicians and appreciative listeners.... 
 The working musician definitely takes over at a certain point the raw 
 materials of his art from the real psychical processes of hearing, 
 inaccessible in full to observation, and then proceeds to construct 
 from them vast new realms without consulting anything that lies beyond 
 the ken of observation." In this volume I think I have succeeded in 
 carrying the psychological groundwork of the previous one forward so 
 as to bridge the gulf between the psychological elements and processes 
 of music on the one hand and on the other the sensory stuff and functions 
 of music as the musician observes them. If my results and analysis 
 are valid, the musician should now have a nearly complete and sure 
 basis to work upon, that will give a scientific foundation to all his 
 elementary observations and satisfy him with a sense of firm ground 
 upon which to build. 
 
 The work of carrying the psychological analysis thus far has not 
 been light. For as things have been till now, the probability of any 
 one person being equally and fully conversant with the science of 
 psychology and with musical history and theory was exceedingly small. 
 I am aware of the great deficiencies in my own preparedness for the 
 latter half of this great double task. I have tried to make up for want 
 of experience by keeping closely in touch with the general trend of 
 the judgments of ripened masters in musical theory of the empirical 
 analytical order. I feel sure that results have offered themselves to my
 
 x PREFACE 
 
 hand that theirs passed by, however closely. And I am certain that 
 they will not be loth to recognise the validity and usefulness of these 
 results. I only wish that they would feel impelled to make themselves 
 familiar enough with the results of psychological work on sound to 
 carry the new and promising basis of theory far forwards into their 
 own fields. In order to facilitate this junction and continuity of work 
 I have ventured as far out into musical regions as the material accessible 
 to me will allow. Analytic musicians will certainly be able to carry 
 things much farther on until the new outlook permeates the whole 
 theory of their art. 
 
 There is still much to be done. Although I feel a growing assurance 
 that the lines of analysis I have followed lead in the right direction, 
 I still feel, not only that the elements of analysis are not nearly well 
 enough assimilated to one another, but also that a much more detailed 
 and exact foundation is required for their final acceptance than the 
 empirical generalisations of writers on harmony afford, however correct 
 and worthy these may be. A better form of evidence is required; and 
 the best I can imagine would be a statistical study of the great composers, 
 seconded by systematic experiments with the best observers. 
 
 In order to bring my results the more quickly into touch with 
 practical issues and exposition, I have again ventured upon an exposition 
 of results in the most familiar terms of elementary musical knowledge. 
 The underlying idea of this account is the construction of a simple 
 framework round which an introductory account of the basis and rules 
 of harmony might be built. 
 
 Some text-books still maintain the effort to link a system of harmony 
 to the traditional lines of scientific explanation founded upon the 
 harmonic partial tones of musical sounds. Of this E. Prout's splendid 
 treatise is in its earlier form a most notable example. But the effort 
 has been a complete failure, whose only result must have been to 
 bewilder any student of a logical turn of mind. Many other writers 
 have abandoned every sort of scientific introduction; and Prout has 
 followed them in the later edition of his work. That course seems, on 
 the whole, preferable to the former. But the bald exposition of analytic 
 generalisations, however true to the great masters they may be, can 
 never be enough. The mind craves for some logical nexus to give the 
 whole mass spontaneous life. Even analytic exposition can never be 
 complete until it has developed into a logical system whose fundaments 
 bear the higher refinements as a tree bears its fruit. And these refine-
 
 PREFACE xi 
 
 ments can hardly be properly approached and stated until the system 
 inherent in the products of analysis has been discovered. 
 
 I hope the lines of introduction I suggest will help to relieve the 
 beginning students whose musical mind is not good enough to be a 
 test for every effect and a storehouse for every impression, of the 
 perplexity and confusion that so soon overwhelms his first efforts at 
 'harmony.' And perhaps the teacher will find comfort and stimulus 
 in having (I hope) a good explanation to offer, by which the student's 
 logical mind may be brought to the support of his natural gifts of ear. 
 
 It may seem strange to suggest that the exposition of the theory of 
 harmony has hitherto been devoid of system. There has certainly been 
 no lack of desire for system and of effort to form it. Dr Shirlaw has 
 recently given us a lengthy account of "the chief systems of harmony 
 from Rameau to the present day." These are very numerous and of 
 the greatest diversity. But it is evident from a study of them that 
 they are all mere castles in the air, as it were. They lack, one and all, 
 any proper sort of foundation. Each man's construction is like a toy 
 castle built upon his outstretched hand. The parts are in the main 
 merely laid down beside and upon one another. The weight of the upper 
 parts repeatedly pulls out the joints from their places; for they are 
 not bound by any mortar. And the builder's own care and anxiety 
 does the most to precipitate the fall of the pile. Perhaps in the end 
 he throws the whole game into the fireplace and warms himself by it 
 while he reflects on what useful thing he might do instead. 
 
 Dr Shirlaw, it is true, in spite of the extensive criticism he bestows 
 upon these theorists, still believes that a good foundation for musical 
 theory is to be found in the "resonance of the sonorous body." "As 
 the sounds of this harmony are contained in the resonance of musical 
 sound itself, all harmony has its source in a single musical sound" 
 (60, 481). But the day is well past in which a system can hope for a 
 moment's success that rests upon such philosophical naivety. The 
 scepticism of Prout and others is far more worthy and hopeful. 
 
 The sonorous body is out of the question. Between it and music 
 there lie not only the phenomenal material of sense but the mechanisms 
 of the ear as well. The ear must stand in real systematic continuity 
 with the physical processes of sound, including as a mere part the musical 
 sonorous body; the phenomena of sense must bear a similar relation 
 to the ear. The problems of musical science consist primarily in showing 
 the system of bonds and relations by which the stuff of auditory sense
 
 xii PREFACE 
 
 is built up into music that delights the soul. There are doubtless various 
 other ways of building up structures of sound than the musical one. 
 But the latter is one of the most important. In it the aesthetic test is 
 applied; a process of selection is set over nature's indiscriminating 
 generosity. But it is obvious that the aesthetic test no more accounts 
 for the auditory material of music than a five shilling entrance fee 
 accounts for the audience that appears at a concert. 
 
 As far as the sensory stuff of music is concerned, then, and apart 
 from such things as rhythm and musical form, no tenable theory either 
 of the auditory basis or of the aesthetics of it has ever yet been advanced. 
 But there does exist a considerable body of generalisations that have 
 been won from the works of the masters by analytical induction. By 
 this I do not mean merely the useful attempts that have been made 
 to form systems of chords; for these have been largely vitiated by the 
 attempt to derive the chords from some fundamental chord or other, 
 when it was evident for strict logical thought and intuition (if one may 
 say so), that all the time no satisfying criterion existed by which any 
 such fundamental chord could be established. D'Alembert expressed 
 this thought in relation to Rameau's efforts when he said they yielded 
 no demonstration, but only a system. I mean rather the body of 
 experience that has been known as the rules of part-writing and of 
 the formation of melody and the factors that modify the operation 
 of these rules. These rules have been the step-children of musical theory, 
 ignored and despised as far as any system building was concerned, 
 treated only as accidents of the all-absorbing science of chords and 
 their origins. But this despised child of musical empiry may well 
 take precedence in musical science and end by being the queen who 
 will show her sisters in music what their functions really are. 
 
 I am far from suggesting that all the main problems have been 
 solved. But it is good that lines begin to appear that seem promising 
 and strong. Many must labour at the task before it will ever be wrought 
 to the satisfaction of all. Still, there seems to be no reason now why 
 rapid progress should not be made, even if no one would be quite so 
 hopeful as Kant was about his efforts as to be sorry for the next genera- 
 tion that would have nothing to do but learn the results. 
 
 When one surveys the actual changes in the fundamental notions 
 of a subject like music required by even very extensive inductions and 
 analyses, they may seem to be surprisingly small. This is perhaps 
 most striking in the problems of consonance and dissonance. It is not 
 so much a striking change of substance that is produced but a far-
 
 PREFACE xiii 
 
 reaching improvement in the body-building capacities of the fundamental 
 nucleus. As cytology shows us, trifling changes in the numbers and 
 arrangement of chromosomes may alter the resulting organism pro- 
 foundly. And ages are required to attain these trifling changes. So in 
 science : some fundamental notions are lethal, they can compound to 
 no living organism; others very slightly different grow and reproduce 
 with great vigour. That I hope may be true of my own analyses. And 
 as great interest attaches to the way in which new notions of promise 
 come to light, I have not sought to obliterate the traces of this process 
 in my writing. A straightforward factual and logical development 
 naturally expunges all these things. But that method really takes 
 much for granted and is only to be used when men are already well 
 disposed to be convinced by the established ways of a science. 
 
 I am indebted to Prof. W. B. Stevenson for a summary of 
 de PearsalFs pamphlet in the British Museum, and to my brother, 
 Rev. T. M. Watt, for reading the proofs. 
 
 H. J. W. 
 
 30th March, 1918.
 
 CONTENTS 
 
 PAGE 
 
 PREFACE . vii 
 
 CHAP. 
 
 I. The reduction of instrumental tones to a single series of pure tones 1 
 
 II. Analytic description and theory of the series of pure tones . . 5 
 
 III. Degrees and theory of consonance and dissonance (fusion) . . 15 
 
 IV. The relations of fusion to beats, partials, and difference-tones . . 23 
 
 V. The consonance of successive tones . . . . . . 31 
 
 VI. The nature of interval 36 
 
 VII. The musical range of pitch 45 
 
 VIII. Our point of view towards the auditory field .... 48 
 IX. The relative importance of synthesis and of analysis ... 55 
 
 X. The equivalence of octaves 65 
 
 XI. Consecutive fifths 81 
 
 XII. The system of facts regarding consecutives 98 
 
 XIII. The reason for the prohibition of consecutives .... 109 
 
 XIV. Exceptions to the prohibition of consecutives . . . .115 
 
 XV. Hidden octaves and fifths, etc. 122 
 
 XVI. A fourth from the bass 133 
 
 XVII. Common chords or concordance 141 
 
 XVIII. Melodic motion in relation to degrees of consonance . . . 151 
 XIX. Melody (or paraphony) as the primary basis of music . . .160 
 
 XX. The factors that modify paraphony 169 
 
 XXI. Retrospect and the outlook for theory 185 
 
 XXII. Synopsis or outlines of instruction 198 
 
 XXIII. The objectivity of beauty 214 
 
 XXIV. Aesthetics as a pure science 221 
 
 WORKS CITED 233 
 
 INDEX OF AUTHORS 236 
 
 INDEX OF SUBJECTS .... 238
 
 We shall now proceed to the consideration of Harmonic and its parts. It is to be 
 observed that in general the subject of our study is the question: In melody of every 
 kind what are the natural laws according to which the voice in ascending or descending 
 places the intervals? For we hold that the voice follows a natural law in its motion, 
 and does not place the intervals at random. And of our answers we endeavour to 
 supply proofs that will be in agreement with the phenomena in this unlike our 
 predecessors. For some of these introduced extraneous reasoning, and, rejecting 
 the senses as inaccurate, fabricated rational principles, asserting that height and 
 depth of pitch consist in certain numerical ratios and relative rates of vibration a 
 theory utterly extraneous to the subject and quite at variance with the phenomena: 
 while others, dispensing with reason and demonstration, confined themselves to 
 isolated dogmatic statements, not being successful either in their enumeration of 
 the mere phenomena. It is our endeavour that the principles which we assume 
 shall without exception be evident to those who understand music, and that we 
 shall advance to our conclusions by strict demonstration. 
 
 ARISTOXENUS (cf. 1, isst.). 
 
 Symphonic are those in which, when they are simultaneously struck or blown on the 
 flute, the melos of the lower in relation to the higher or conversely is always the 
 same or (in which), as it were, a fusion in the performance of two tones occurs and 
 a kind of unity results. 
 
 Diaphonic are those in which, when they are simultaneously struck or blown, nothing 
 of the melos of the lower in relation to the higher or conversely appears to be the 
 same or which show no sort of fusion in relation to one another. 
 
 Paraphonic are those that, standing in the middle between the symphonic and the 
 diaphonic, yet appear symphonic when played [on instruments, or "in heterophonic 
 passages on instruments" (11, 139)]. 
 
 GAUDENTITTS (cf. 54, 69).
 
 CHAPTER I 
 
 THE REDUCTION OF INSTRUMENTAL TONES TO A SINGLE 
 SERIES OF PURE TONES 
 
 THE world of sound is bounded by the two extremes of pure tone and 
 mere noise. The home of music lies in the lands around the ideal of 
 tone. This ideal forms the first problem of musical science. 
 
 It is properly termed an ideal because pure tones rarely, if ever, 
 occur under natural circumstances. That is evident from the familiar 
 fact that, however perfectly a musical instrument may be played, its 
 tones are easily distinguishable from those of other instruments, even 
 though they may be of the same pitch. The same series of tones of 
 exactly the same pitches, e.g. the diatonic scale on a c 1 of 264 vibrations 
 per second, may be given by an indefinite number of instruments, and 
 will be recognised as different on each, in spite of the sameness that is 
 obviously common to all. 
 
 This peculiar complexity of tone has been explained by modern 
 research in a way that at least in principle is complete and final. How- 
 ever pure and beautiful an instrumental tone may be, it can be analysed 
 into audible parts by special means of two kinds. In the first the ear 
 is provided with instruments which will increase the intensity of certain 
 parts of the tone if they are present in the tone to be analysed 
 beyond that of the other parts. When the resonator is placed against 
 the ear, it seems to be full of the magnified sound, whose pitch may 
 be surprisingly different from that of the tone it comes from. But its 
 presence in the resonator is easily shown to depend upon the studied 
 tone. Whenever the resonator is placed against the ear, it appears; 
 and it only appears in the resonator when the tone being analysed is 
 sounded, unless, of course, it is given from some other source at the 
 same time. Any such experimental error can be easily avoided in most 
 cases. After a little practice with the resonator, the partial tone will 
 often be distinguishable in the whole tone even when the resonator has 
 not been placed upon the ear. This is not the result of imagination or 
 illusion. It only means that the ear has now been trained for this 
 particular case to expect a certain partial tone and to direct observation 
 
 w. v. M. 1
 
 2 REDUCTION OF INSTRUMENTAL TONES [CH. 
 
 specially upon it, so that it appears to be more or less abstracted from 
 the whole. 
 
 A generalisation of this procedure gives the second method of analysis. 
 The ear is first prepared for special observation (or abstraction) by 
 listening for some time to the tone expected to occur. The tone to be 
 analysed is then presented and if it contains the prepared tone as a 
 partial, the latter will probably be heard sounding faintly through the 
 whole. If the ear has been prepared for a tone whose pitch is not quite 
 the same as that of the partial actually present in the whole, the listener 
 will not hear the pitch he expects, but another that lies in pitch near the 
 one expected. When the ear has thus been trained for many or for all 
 the partials of a tone, it may be able to run through them all in sequence 
 without any special preparation. And in the course of time it may learn 
 to do this for any sound of a tonal nature. Even then, however, the 
 tones of well played musical instruments do not cease to be the 
 beautifully perfect unities they were before. They do not fall to pieces, 
 as it were, permanently, but only when the attention is concentrated 
 and moved from one of the partials to another. 
 
 The occurrence of these partials is due to the fact that most musical 
 instruments when brought into a certain rate of vibration n times 
 per second, fall at the same time into various rates of vibrations that 
 may be any whole multiple of n : 2n, 3n, 4n, 5n, etc. The pitches 
 corresponding to these ratios of vibration are : octave, octave and fifth, 
 double octave, double octave and third, double octave and fifth (6n), a 
 pitch slightly flatter than the $ above (In), the triple octave (8n), then 
 the d (%ri) and the e (lOn) above this c, and so on. The pitch of any 
 partial may easily be reckoned out from a knowledge of the ratios for 
 the chromatic scale and by approximations thereto. These ratios are : 
 
 c, $, d, #, e, f, /3, g, o>, a, &, b, c 1 
 24, 27, 30, 32, 36, 40, 45, 48 
 
 i' it . I t t. * It t t t V i 
 
 All of these ratios can be derived from those of the octave (2 : 1), 
 fifth (3 : 2), and major third (5 : 4) that have played so important a 
 part in the history of musical theory. 
 
 The existence of partial tones has been confirmed objectively in a 
 number of ways. In various cases the presence of vibrations corre- 
 sponding to the pitch of the partials heard can be demonstrated to 
 vision. A long stretched string may be seen to vibrate, not only in its 
 whole length, but also in parts of such length as would give the various
 
 i] TO A SINGLE SERIES OF PURE TONES 3 
 
 partials as independent tones, if these parts of the string were made to 
 vibrate separately from the rest of the string (cf. 35, 91 f.). In other 
 cases the motion of a minute mirror standing in connexion with a 
 vibrating membrane may be photographed with so little error that the 
 result may be taken as representing the motions of the air that excite 
 the vibrating membrane (40, 78 ff.). (Phonography is dependent upon 
 such a vibrating membrane, and everyone is aware how good a repro- 
 duction of sound may be obtained thereby.) When the photographs so 
 obtained are subjected to mechanical 'harmonic analysis,' the partial 
 tones that result correspond very closely with those that can be heard 
 by the most careful analysis with resonators or with prepared attention. 
 
 Such studies as these show that tones of the same nominal pitch 
 from different instruments differ only in respect of the group of partials 
 from the full series (of possible multiples of ri) that they contain and in 
 the relative strength of these. Some tones like those of tuning-forks 
 and of the flute contain very few partials, perhaps only the first. Others, 
 such as pianoforte tones, are rich in the lower partials. Others again, 
 like those of the trumpets, contain a host of high partials in great 
 strength, which give them their peculiar brightness and brilliancy. And 
 so on. The results of this line of study will be found extensively in 
 special treatises and in text-books of physics (40, 175 ff.; 20, H8f., etc.). 
 
 Partials may be eliminated from the tones of any instrument by 
 the method of physical interference. That consists in principle of the 
 conduction of a sound containing at least one partial other than the 
 fundamental component of the tone, along a tube which for a certain 
 length is double and then unites again to enter the ear. The one doubled 
 part is made longer than the other by half a wave-length of the partial 
 to be eliminated. When the parts unite, each will bring this partial in 
 exactly opposite phase to the other. If the one is at the phase of 
 maximal condensation of the air, the other will be at that of maximal 
 rarefaction; and the result will be the elimination of that component 
 of the aerial disturbance. If there are many partials in the tone, there 
 will have to be a special device of this kind for the elimination of each, 
 so that the apparatus for the production of a pure tone from an instru- 
 mental one is apt to be somewhat complicated, unless special care is 
 taken to begin with a tone containing very few partials, such as that 
 of a tuning-fork. 
 
 In spite of the difficulties that thus face any complete generalisation, 
 no one doubts for a moment that the series of perfectly pure tones, each 
 consisting of a fundamental with no upper partials at all, thus isolated, 
 
 12
 
 4 REDUCTION OF INSTRUMENTAL TONES [CH. i 
 
 is one and the same, no matter from what instrument it may have 
 been derived. This is in perfect accordance with the results of the 
 attentional analysis of instrumental tones. For the partials separated 
 by the attention do not seem to differ from the (pure) tones of identical 
 pitch, otherwise produced, in any such way as would make us believe 
 that the series of pure tones is not the same whatever its source may be. 
 Thus we obtain a simpler starting-point for our study of tone the 
 series of pure tones, each one of which corresponds to a certain fixed 
 rate of aerial vibrations, unmixed with any other rate. Now this series 
 is perfectly continuous. If we start from any ordinary pitch, e.g. middle c 
 of 264 vibrations per second, we can raise or lower the pitch of tone 
 gradually, producing differences as minute as the mechanical means at 
 our disposal will allow. There is no reason in the nature of tone why 
 we should select any one pitch or rate of vibration for our c or a rather 
 than any other. And even when a standard pitch has been adopted 
 for practical purposes, minor variations due to change of temperature, 
 mis- tuning, etc., are inevitable. It has been claimed that the vibratory 
 rate of 256 should be taken as the standard of 'philosophic pitch,' 
 because 256 = 2 8 ; i.e. if we imagine a tone of one vibration per second, 
 the (fictional) tone of two vibrations would be the octave of it, four 
 vibrations would give the double octave, and so on, so that the eighth 
 octave would give us 256 vibrations (50, 33 ff.). It is certainly very 
 useful to have a commonly accepted standard for convenience of 
 reference. Then we know what rate of vibration is implied by any 
 nominal pitch, e.g. A I} without having to give it separately. But the 
 standard now perhaps most commonly in use is a c 1 of 264 vibrations 
 per second. One advantage of this basis (although a slight one), is 
 that it is a multiple of 24, and so can be readily used in connexion 
 with the diatonic series of ratios stated above. I shall use this standard 
 throughout the following pages unless some other standard is specially 
 indicated. The usually current nomenclature of octaves may be looked 
 upon as starting from 'middle c,' the c common to the baritone and 
 contralto voice, which is called c 1 . Above that the octaves are c 2 , c 3 , c 4 , 
 c 5 (the highest note on the large concert grand piano), c 6 , etc. ; below 
 we have c, C, Cj, C 2 (A 2 is the lowest note on the same instrument). 
 Plain letters will thus indicate absolute pitch, italicised letters relative 
 pitch.
 
 CHAPTER II 
 
 ANALYTIC DESCRIPTION AND THEORY OF THE SERIES 
 OF PURE TONES 
 
 HAVING thjis reduced the usual tonal material of music to its simplest 
 components, we have now to describe this continuous series. The terms 
 of our description, to be scientifically useful and explanatory, must be 
 such as will bring tones into systematic connexion with as many other 
 similar objects as possible. 
 
 Being dependent upon the working of a sense organ the cochlea 
 of the ear, tones are classified in psychology as sensations. We 
 naturally expect them to show great similarity to the sensations we 
 get from our other sense-organs, such as the eye (vision), the tongue 
 (taste), the skin (touch, temperature, pain), and various others, such 
 as hunger and thirst. The similarity of all these to one another is certainly 
 not at first striking. And it has usually been thought that the differences 
 are far more numerous and important than any resemblances there 
 may happen to be. Many men, indeed, judging by the perennial failure 
 of the attempt to bring our different senses into systematic connexion, 
 have adopted a standpoint of extreme scepticism towards any such 
 claim or expectation. But since psychology became, some decades 
 ago, an experimental science, the study of the sensations has been 
 pursued most carefully and exhaustively, and the real relations of 
 resemblance and of structure between our various sensations have 
 gradually grown clearer. Beyond this nothing is required but a frank 
 and determined rejection of the old prejudice and a whole hearted 
 effort to work out the inner similarity of sound and of the few other 
 senses we have. The problem then is to describe the tonal series so as 
 to show the inner connexion not only between all the parts of the 
 series, but between tones and the sensory objects of the other senses. 
 
 The first and most obvious feature of sounds is that which distin- 
 guishes them from the sensations of other senses. No kind of sound or 
 group of sounds is ever confused with a sight or with a touch. Psycho- 
 logists call this a difference of quality. The word quality is often used 
 by musicians to designate that difference of tones of the same pitch 
 which is due to the peculiar blend of partials they contain. It is better, 
 however, to call this the (pitch) blend of tones. For practical purposes
 
 6 ANALYTIC DESCRIPTION AND THEORY [OH. 
 
 that word is the best which most readily suggests the thing named, 
 or its cause, or the like. The word blend is in common use as a name 
 for similar differences in objects that appeal to other senses, especially 
 to taste and smell, of which the latter is the more important. The blend 
 is here due to the mixture of the components. Similarly the blend of 
 a tone will be the difference due to the admixture of partials, which a 
 trained ear can learn to pick out and name, as a practised palate will 
 detect the components of a tea or the varying flavours of a wine. This 
 word 'blend' seems better than the French word 'timbre,' which does 
 not fit into our language either in its native pronunciation or in ours. 
 
 The second attribute that is found in all tones and in the sensations 
 of other senses is intensity. Both scientific and popular usage agree 
 as to the meaning of this term. The word loudness is not so useful for 
 classification, because it is inapplicable to the other senses and so does 
 not serve to indicate any variant common to them all. One word of 
 frequent occurrence must be carefully avoided in this connexion, 
 namely, volume. We think of loudness as great volume when many 
 instruments sound together as in an orchestra and so make a very intense 
 sound or a mass of very many sounds. Having thus associated many 
 sounds with much sound, we often use the word where there is obviously 
 only one sound present, as when we speak of the great volume of a 
 singer's voice, especially of a contralto's or a bass's. Here a touch of 
 the scientific usage begins to appear. But that highly justifiable usage 
 does not tolerate any confusion of volume with loudness. 
 
 Volume is properly used to distinguish that difference between 
 tones of different pitch that makes the low tone great, massive, all- 
 pervasive, and the high tone small, thin, and light. The other words 
 we use to designate differences of pitch have the same sort of association. 
 Sharp and flat are closely akin to thin and broad, or small and large. 
 The Latin and French words gravis, grave, acutus, aigu, bear the same 
 implications. In the eighth Problem on Music Aristotle asked: "Why 
 does the low tone dominate the higher? Is it because the low is the 
 greater? For it is like the obtuse angle, while the other resembles the 
 acute angle." And the twelfth Problem answers: "Is it because the 
 low tone is great and therefore more powerful and because the small 
 is included in the great? " (65, 17, 19; cf. 16, 13, 19). 
 
 Although the words of these sentences are very suggestive to a 
 theorist of the present day, it is doubtful whether Aristotle meant really 
 to ascribe differences of size to the tones as mere sounds or sensations. 
 His mind was very much impressed by the discovery that the low string
 
 n] OF THE SERIES OF PURE TONES 7 
 
 gives out not only its own tone but the higher octave, so that, as we 
 should say now also, the low tone contains 'the higher one' its own 
 first higher partial, the octave of itself. But Aristoxenus refers to "the 
 blunder of Lasus and some of the school of Epigonus, who attribute 
 breadth to tones" (1, 167). And many modern writers have inclined 
 more or less tentatively towards this idea as an explicit description of 
 tones as such. We must now certainly take the idea with complete 
 seriousness and think of tones as of different size or mass or bulk, just 
 as a visual sensation can be of different size in respect of its mass or 
 area, or as pain and hunger can be large and massive, or as pain and 
 touch can be small as sand or needles. There is every reason to believe 
 that this difference of volume or extent is dependent upon the number 
 of elementary sense-organs of hearing that are in action at the same 
 time. But that is a question for physiology. 
 
 The only other property of pure tones is what we commonly call their 
 pitch. By pitch tones fall into a definite order or series. This is not 
 naturally a discrete series : like the ordinal numbers, of which each 
 one is an individual separated from the next by a unit of space, into 
 which other numbers of a fractional nature may be fitted; or like the 
 pitches of the diatonic scale. It is a continuous series : we can pass 
 from any one point of it to any other by gradations that are not dis- 
 tinguishable from those that lie next to them on either side, but that 
 are distinguishable from those that lie a certain distance away on either 
 side. And the series is ordinal, not because it can be considered con- 
 ceptually as a continuous series of positions, but because it appears 
 so to us phenomenally, as is often said, or merely as sensation. The 
 series of colours of the spectrum, merely as colours (i.e. apart from their 
 position in the dispersed spectrum, and from the wave-length they 
 depend upon) can be treated in an ordinal way, although that series 
 is, as sensation, really a series of qualities. But the pitch series is, as 
 sensation, itself really ordinal 1 . It presents itself to us as ordinal and 
 
 1 Aristoxenus wrote: "Tension is the continuous transition of the voice from a lower 
 position [r6irov~\ to a higher," etc. (1, 102. 172; 14, sat.). The Greek term is highly suggestive. 
 But it does not seem certain that he meant by it more than cessation of change of 
 the voice, a permanence of one kind of activity. It is, of course, significant that in this 
 case we naturally incline towards the term 'place' or to the idea of the voice's 'moving.' 
 We do not incline to say that the weather ' moves,' or the colours of leaves ' move ' in the 
 autumn, when we mean only that they change. No doubt Aristoxenus used the terms 
 ' place ' and ' motion ' at the suggestion of the ordinal and motional aspects of tone. Never- 
 theless his concepts of position and motion of the voice probably did not include more 
 than what he might have attributed to a thing that only changes, i.e. progress of change 
 and arrest of change Cf. 80, 221 1.
 
 8 ANALYTIC DESCRIPTION AND THEORY [CH. 
 
 calls for ordinal names, whether we know anything about the wave- 
 lengths that cause it or not. 
 
 The attributes of tones thus far enumerated are: quality, intensity, 
 volume and pitch. The relations between these four are an important 
 problem. It has been suggested that intensity is a sort of density of 
 sensation, as it were. Just as a gas may fill a certain volume and yet 
 be very thinly scattered throughout it, so it is thought a sensation 
 may be of one and the same quality, and volume, and pitch, and yet 
 be more or less dense or intense. Suggestive arguments in favour of 
 this view have been advanced, but they do not yet seem sufficient for 
 their purpose. 
 
 The relation between pitch and volume is much clearer. When 
 tones are compared with noises, a marked difference is apparent. Tones, 
 as everyone feels and knows, are smooth and regular, noises are rough 
 and irregular. Tones may also be said to be balanced and symmetrical, 
 while noises are chaotic and disorderly. These descriptions obviously 
 refer to the volume of tones, not to their pitches. A pitch has only 
 a definite position or place; it is not smooth or balanced. But pitch 
 gives tone a position as a whole; it is by means of pitch that tones are 
 brought into a definite and accurate series, and their volumes along 
 with them. The question then arises : what position has pitch itself 
 in the tone's volume? 
 
 This is not an absurd question, but a very natural one. For if tones 
 have an aspect of volume and can be arranged in a very definite and 
 single series by means of pitch a property that is distinguishable 
 from volume; and if pitch is not only thus really ordinal, but is also 
 felt as ordinal or appears to us so as a property of sensation; it is 
 perfectly natural to suppose that what is thus ordinal is a part of the 
 tone's volume and to ask in consequence which part of the volume 
 constitutes the pitch? 
 
 Of course, one's habits of thought may oppose this line of inquiry. 
 One of the greatest obstacles to the advance of knowledge is the 
 opposition our minds offer by the mere force of unfamiliarity to the 
 application of old and simple notions to common objects to which they 
 have not hitherto been applied. The mind seems to refuse to establish 
 the desired connexion. All sorts of excuses and objections are offered 
 to the new invitation. "Metaphors are so misleading." But it is not 
 a case of metaphors now. Pitch is no mere analogy; the ordinal status 
 and arrangement of tones is one of the bed-rock facts of music. And
 
 n] OF THE SERIES OF PUKE TONES 9 
 
 ' volume ' is no more a mere simile than is interval or concord or discord . 
 It is as much there as any fact could possibly be. Psychologists admit 
 it more and more frequently, and it is only a matter of time till everyone 
 who considers the subject will agree with them. Nor is it 'mystical' to 
 suggest that pitch has a position in volume. A line of well-founded and 
 logical thought is only mystical to those who do not take the trouble 
 to follow it carefully. A mystic is one who claims to have special insight 
 or experience which he has discovered by accident or providential 
 good-will and which he is powerless to reveal to others either because 
 it defies all description or because, not knowing how he himself attained 
 it, he is unable to lead thither all who would share it with him. But 
 there is nothing mystical about pitch or tonal volume; nor are the 
 ordinary logical processes of inference held to be the special privilege 
 of a few minds. 
 
 We may therefore consider our question clear and reasonable. And 
 the most likely answer follows naturally from the apparent balance 
 and symmetry of tones as compared with noises. We may assume that 
 pitch holds a central position in volume. And, as pitch is ordinal, while 
 volume suggests a volume of parts or particles, we may go on to assume 
 that pitch is constituted by a specially prominent or noticeable part 
 of the volume of sound that makes up a tone. We certainly do not 
 hear tone as a group of distinguishable particles like a handful of sand. 
 We hear it as a continuously smooth closed volume. But nevertheless 
 a part of this whole volume might well be more noticeable than the 
 rest, just as a part of a variably 'toned' visual surface may be most 
 deeply coloured red, for example, although we could not pick out 
 and isolate any part of it that would be all, and nothing less or more 
 than all, the reddest part. Yet no one doubts that a visual surface 
 consists of a mass of minimal particles of colour surface, grouped into 
 a continuous whole. These particles are presumably the minimal areas 
 of colours given by single recipient visual organs the cones (and rods) 
 of the retina. There are also in the ear elementary receptors of sound; 
 and these presumably afford us the minimal particles of sound that 
 make up tone. 
 
 Now all (pure) tones are the same in symmetry and balance and 
 smoothness. So we may consider this central position and predominance 
 of pitch to be characteristic of pure tone as against all grades of noise, 
 which are relatively rough and unbalanced, vague or indefinite in pitch 
 or marked by many prominent points of pitch. Tones differ from one 
 another in size of volume and in the ordinal position of their pitches
 
 10 
 
 ANALYTIC DESCRIPTION AND THEORY 
 
 [CH. 
 
 relatively to one another. The pitch of a higher tone lies a little to one 
 side of the pitch of a tone just lower in pitch; and the pitches of all 
 tones together form a single linear series, having the tone of greatest 
 volume at one end and the tone of least volume at the other. If we 
 were to project the volumes and the pitches of all the tones of the series 
 against one another in our thought, we should obtain a scheme of the 
 following kind: 
 
 High Tones j 
 
 Low Tones 
 
 P 
 
 Pig. 1 
 
 P' 
 
 If we placed all the pitch-points on a perpendicular line above one 
 another, we should indeed represent the decrease of volume (as we go 
 up) properly by the decrease in the breadth of the line used, while the 
 symmetry and balance of tone would be indicated by the central position 
 of the P point in the volume line (VI 'lower' end of volume, 
 Vh = 'higher' end of volume). But we should not have given any 
 representation of the fact that the pitch of a tone higher than another 
 lies on one side of the pitch of the latter in an ordinal series. This series 
 is quite properly indicated in our figure. 
 
 We have as yet no proof for the assumption that the Vh ends of 
 all the volumes should lie perpendicularly above one another, or 
 whether in mere projection on the base line of the figure (which may 
 be supposed to represent the greatest possible volume or the lowest 
 possible tone), or in reality should be the same point. It is conceivable 
 that the pyramid of tones should be acute or obtuse angled rather 
 than right-angled. But these alternatives are far from likely for various 
 reasons of which the most important will be set forth immediately.
 
 n] OF THE SERIES OF PURE TONES 11 
 
 It would follow from our scheme that if the projected series of pitches 
 is in any way real, that ought to appear in an unmistakable manner 
 when tones of different pitches are given simultaneously. And this is 
 the case. Simultaneous tones seem to be mixed together or to fuse 
 with one another or to intermingle. They never appear to be entirely 
 apart from one another as two patches of colour often do, when they 
 are separated by a patch of a third colour or when they just bound 
 one another. Each patch of colour is seen as well when both are given 
 together as when they appear singly and successively. Not so two 
 tones; they always appear to be in each other's way, to crowd upon 
 or to overlap one another. This holds even for the greatest extremes, 
 when the highest and lowest tones are given together. And yet at the 
 same time the two tones by no means completely lose their individuality 
 or become indistinguishable in this 'mixture,' as two colours do when 
 they are mixed by being cast upon the same surface or by the rotation 
 of them on a disc. Blending of colours gives a new colour in which the 
 components are essentially indistinguishable by any one who did not 
 see them before their mixture. But musical folks can detect the com- 
 ponent tones of a chord with ease and certainty. In spite of the over- 
 lapping or interpenetration of tones they are more or less readily 
 distinguishable. Their pitches generally strike us as being the same in 
 mixture as in isolation. 
 
 If, as we have supposed, the series of pitches represents a series 
 of real particles of sound, a ready explanation of this peculiar inter- 
 mingling of tones is at hand. Then the sounds that form the highest 
 musical tone will form the last (or highest) part of the volume of all 
 simultaneous lower tones. Being the same sounds, the two common 
 parts will overlap or intermingle; but, as the higher tone adds its 
 intensity to the high part of the low tone, and especially the predominant 
 intensity of its pitch, the high tone will stand forth, and be detectable 
 in the lower tone in spite of the overlapping of the two. And the 
 rectangular shape of the scheme we have figured is justified. The 
 possibility of an obtuse-angled figure is then excluded; for that would 
 imply that the volume of the highest tone lay beyond the volume of 
 the lowest tone, so that a medium tone would not mix with, or obscure, 
 a very high tone in the slightest degree. And the probability of an 
 acute-angled form being the true scheme is equally small; for in that 
 case the movement of pitch that accompanies the continuous decrease 
 of volume would proceed, as it actually does, steadily in one direction, 
 but only up to a certain point, after which the direction would be
 
 12 ANALYTIC DESCRIPTION AND THEORY [CH. 
 
 reversed. Of this there is no actual trace in hearing. We may therefore 
 for the present safely follow the scheme depicted in the figure. We shall 
 obtain further evidence on this point when we come to the study of 
 the grades of fusion. 
 
 We conclude, therefore, that tone is a mass or volume of minute 
 (hypothetical) particles of sound sensation, of which those at its centre 
 are the most intense, while the others grade themselves on either side 
 in the whole volume so that the mass appears regular and balanced. 
 Since in a pure tone no other predominating points or pitches appear, 
 we must suppose that the intensity of its constituent particles decreases 
 gradually and regularly from the central pitch towards the limits of 
 the volume 1 . In this respect every tone is appreciably the same, no 
 matter what its pitch may be. The particles that constitute volume 
 must be called hypothetical, because the volume does not appear to 
 be a group of distinguishable parts. We do not separately experience 
 the minimal particle except possibly in the case of the tone or sound 
 of the smallest possible volume, i.e. the highest audible sound. That 
 may be approximately the minimal particle. Between this highest 
 particle (or pitch) and the pitch of any tone there lies the whole series 
 of pitches that leads from the latter to the former. And we have reason 
 to infer that half of the volume of any tone is made up of the pitch 
 particles that appear in the series of tones progressively higher than 
 itself up to the highest tone. So the lowest audible tone would consist 
 in its one ('upper') half of the whole series of audible pitch-points. 
 The other (or 'lower') half of its volume is not used for the formation 
 of the pitches of other tones. That is, of course, no argument against 
 its existence. 
 
 Every tone must, therefore, contain within it the volumes of any 
 higher tone. And all tones are constituted from a single series of sound 
 particles, of which they incorporate a series always beginning at the 
 common upper end and stretching downwards as far as the size of each 
 tone's volume requires. These and other such relations can easily be 
 read from the figure already given. 
 
 1 Wm. Gardiner in his popular compendium of musical topics, The Music of Nature 
 (13, 188) gives a curious diagram entitled: "The wind instruments the shape and order 
 of their tones from the lowest to the highest," which shows a column of twenty-one oval 
 figures, coloured differently for the different instruments, with a small dot at the centre 
 of each. These ovals grow gradually smaller towards the top of the column. They might 
 all be inscribed in an angle whose sides were about five centimetres long and two-thirds 
 of a centimetre apart at the ends. Gardiner did not in any way elucidate in words the 
 reasons that led to this diagram.
 
 ii] OF THE SERIES OF PURE TONES 13 
 
 The series of pitches may be said to define clearly one-half of one 
 of the dimensions of tonal volume. We say 'one,' because we do not 
 feel tone to be a mere line or length, but a volume; something areal 
 or massive or round, as it were. Of course, in so far as we think of tones 
 from the pitch point of view they fall into a perfectly definite ordinal 
 or linear series of no thickness at all, so to speak. Yet when we look 
 upon tone naively as a whole, it is as mass or volume that it appears 
 before us. This implies that it has at least one other dimension, different 
 from that indicated by pitches. The musical aspects of tone give us 
 no means of demonstrating or of defining this direction. For these 
 aspects are concerned only with the variations of tone in pitch and 
 volume, i.e. only with longitudinal variations. And no transverse 
 variation accompanies these. We must turn to quite a different function 
 of hearing, one that is dependent not on either ear separately or equally, 
 but on both ears integratively or in a combined purpose. This function 
 enables us to localise sounds towards the right or the left ear and in an 
 imperfect way round the head in space. A careful study of binaural 
 hearing leads to the conclusion that every tone has breadth as well 
 as length. This breadth is also marked out into a series by the variations 
 of binaural localisations or 'local signs'; and, unlike the pitch series, 
 this one is traversable as far towards the one end ('opposite the one ear') 
 as towards the other ('opposite the other ear'). 
 
 We have no means of comparing or of measuring the extents of the 
 two series with one another, such as superposition or the like. But we 
 do not feel this ignorance or disability as a difficulty or mystery; for 
 we simply do not think, we have no inclination to think, of these series 
 in relation to one another. On the contrary, the one is the musical 
 aspect of tone, the other is the basis of its spatial aspect. Music is the 
 same for a person of normal hearing whether it is played to the right 
 or to the left of him. And the tonal nature of a warning signal is largely 
 insignificant, if we but gauge the position of its source aright. These 
 two interests of sound appeal to very different practical functions in 
 spite of the close relation of their ultimate bases to one another. Besides, 
 the total transverse breadth of tone probably hardly ever varies, unless 
 in passing from purely uniaural to binaural hearing. It is only the 
 point of emphasis in the total breadth that varies and so forms a basis 
 for localisation towards the one ear or the other. The longitudinal or 
 musical aspects of tone would, therefore, be unaffected in any way. 
 And in case there might be any diversion of interest, we neither en- 
 courage our musicians to rove around us while performing, nor do we
 
 14 DESCRIPTION AND THEORY OF PURE TONES [OH. n 
 
 practise an oscillation between hearing in the usual way and hearing 
 with one ear only. But even though the breadth of tone is not, and 
 from the nature of the case cannot be, a musical variant, it is still there 
 all the time in musical sound, and doubtless makes it what it is a 
 volume. (For proof of this transverse aspect of sound, see 77, Chap, ix.) 
 
 We may sum up our conclusions by saying that pure tone is a volume 
 of sound in which a minute part or point is most intense, while the 
 rest is graded smoothly and symmetrically around this probably 
 central part, the pitch of the tone. By means of pitch, tones can be 
 arranged in a series of diminishing volumes. But no two simultaneous 
 tones fail to fuse or blend with one another or to be heard through 
 one another. We therefore infer that the volume of any tone includes 
 not only the pitch, but the whole volume of every higher tone; so that 
 all tones may be reduced to a single series of hypothetical particles 
 of sound, one half of which series we actually hear as the pitches of 
 tones. Apart from these pitch points the nearest approach to the 
 hypothetical particle of sound that is ever separately experienced by 
 us is the minute volume of the highest audible tone or sound. The 
 other dimension of tonal volume appears in the ' local signs ' of binaural 
 hearing. But there is no method whereby we might compare the size 
 or length of this series with the length of the pitch series, so as to say 
 what the exact shape of tonal volume is. We must be content with the 
 close correspondence shown between the feeling of tonal volume and 
 what we have proved it to be. The probable shape of tone is like a 
 visual parallelogram that is longer than it is broad. It varies greatly 
 in its length but probably not, or hardly, at all in its breadth. This 
 relation between breadth and length constitutes the typical and constant 
 form of tonal volume. It may be illustrated diagrammatically thus: 
 
 P 
 
 L c 
 
 L e L 
 
 1 Ve \b 
 
 L 
 
 ! Vc 16 
 
 \. 2. Diagram of two tones a tenth apart, e.g. c 1 and e 2 , showing the constant breadth (b) 
 and the variable lengths (1>-Z/ and Lf-L) of the volumes of the tones ( Vc and Ve), 
 and the intensive differences within the volume (L*), the pitch (p) being the pre- 
 dominant point of the whole.
 
 CHAPTER III 
 
 DEGREES AND THEORY OF CONSONANCE AND 
 DISSONANCE (FUSION) 
 
 IN a general way the various grades of consonance between two tones 
 sounded simultaneously have been long established and are disputed 
 by no one. The ancient Greeks recognised three grades of consonance 
 or symphony the octave, the fifth, and the fourth. To these, in a 
 certain sense 1 , the third (but not the sixth) was added by Garudentius 
 about the end of the third, or the beginning of the fourth, century A.D. 
 (cf. 66, 72). Modern musical theory grades the consonances as perfect 
 octave, fifth, and fourth, and imperfect the major and minor thirds 
 and sixths. All the other intervals smaller than the octave are dis- 
 sonant the sevenths, the tritone (diminished fifth or augmented 
 fourth) and the seconds. The increase of any interval by an octave 
 is not held to make any change in its consonance or dissonance (except, 
 in order to satisfy the needs of certain theories, in certain chords). 
 
 Of the dissonances, however, the minor seventh, as also sometimes 
 the tritone, is commonly considered to be almost consonant. Frequent 
 reference is made (especially by those who look to the series of partials 
 for the causes or origins of chords and scales) to the approximate 
 equality between the diminished seventh and the 'natural' seventh 
 formed between the fourth and the seventh harmonics of any funda- 
 mental (in the case of the tritone the fifth and seventh harmonics). 
 The difference is only a sixty-fourth part in terms of the ratios of 
 vibration of the two partials. It has often been claimed that the chord 
 cegb\), taken exactly in the ratios 4, 5, 6, 7, is really a concord (cf. 
 66, 7i). Those who look for some fundamental development, say of the 
 ear itself, underlying the progress of musical art, might claim this 
 tendency as another step forwards beyond the one first recorded by 
 Gaudentius. Various theorists have even believed that all the grades 
 of consonance have their ground in habit, racial if not individual. For 
 the partial that occurs oftenest and loudest with any fundamental is 
 probably its octave, the next is the fifth above, and so on in the series 
 of partials. What we hear oftenest together, the mind (individually) or 
 
 1 Which we shall have occasion to examine more closely later on.
 
 16 DEGREES AND THEORY [OH. 
 
 the ear (racially) comes to hear as one or at least as an agreeable combina- 
 tion because the ear or the mind has got accustomed to hearing 
 them together. Thus in its progress the art of music is gradually 
 climbing the ladder of partials. We have already accommodated 
 1, 2, 3, 4, 5, and 6 in all possible groupings and we are now absorbing 7. 
 In some future we shall bring even 9, and 11, and 13 and others that 
 we now treat as dissonances, over the border as consonances 1 . One 
 writer has even tried to create an experimental basis for this theory 
 by giving persons much practice in placid attention to dissonances. 
 
 A number of experimental investigations have been made with the 
 purpose of grading the diatonic intervals more accurately than the 
 usage of music indicates. The general result of these yields the series : 
 octave, fifth, fourth, major third, minor third, major sixth, minor 
 sixth, tritone, major second, minor seventh, minor second, major 
 seventh; or in a useful symbolism, which will be maintained in the 
 following pages, 0, 5, 4, III, 3, VI, 6, T, II, 7, 2, VII. This series 
 represents comparatively to what degree the two tones seem to fuse 
 with one another to form a unitary whole. is more fused than 5, 5 than 
 4, and so on. 
 
 In other experiments an attempt has been made to obtain figures 
 representative of the degree of difference between the grades of fusion. 
 Highly trained musical ears might attempt to indicate these quantities 
 by direct estimation. But their judgments are quite open to the influence 
 of their knowledge of musical practice or of any other thoughts or 
 theories they may have. For they recognise the interval as soon as it 
 is sounded and so can think of it whatever occurs to them. And the 
 relative quantity of fusion is not, as it were, marked on the face of 
 intervals. More or less unmusical minds, however, are often quite 
 unable to distinguish the two pitches of an interval, or to recognise 
 one of them or the interval as a whole, etc. Consequently they think 
 nothing about them by way of memory, so that they are sure to be 
 freer from suggestive influences than musical minds. This advantage 
 
 1 Of. A. E. Hull, 22, 265: "The standard of aesthetics varies from age to age. A com- 
 bination of notes which one generation accepts only on sufferance will be received by a 
 later generation with equanimity or even delight: Monteverde's Sevenths, Wagner's 
 Ninths, Gounod's Thirteenths, Debussy's Twelfths, and so on." On page 115 Hull gives 
 a scheme of development which places the first two partials as primeval, the second two 
 as mediaeval, 5, 6, 7, 8 and 9 as of the 18th and 19th centuries, 10 to 16 in the form of 
 e,/$, g, at?, &t?, 6 and c as "whole-tone scale, Debussy, Scriabin." "Undoubtedly Scriabin's 
 exploitation of the higher harmonics will lead to wonderful developments, which are 
 even already in evidence" (p. 271). But he says elsewhere (p. 265), "Many passages 
 in Scriabin's work seem ugly to us, some almos repulsively so." Cf. 24.
 
 m] OF CONSONANCE AND DISSONANCE 17 
 
 weighs against their want of practice and skill, which can even be turned 
 to special account. For the unmusical mind's failure to detect the 
 pitches in intervals may be used to obtain an index of their grades of 
 fusion. The oftener one fails to detect that two sounds have been 
 presented, the more unitary and fused we may suppose the whole sound 
 mass to have been; the oftener the listener feels that two sounds have 
 been given, the less unitary and fused must the combined sound have 
 appeared to him. One series of experiments in this manner gave 80 % 
 of answers asserting the presence of one tone where there had really 
 been an octave ; 50 % where there had been a fifth ; 35 % for the fourth ; 
 30 % for the minor third ; 27 % for the major third ; 23 % for the tritone. 
 These and other results have shown that there is a marked difference 
 between the octave and the fifth, and between the fifth and all the 
 others; but that amongst these last there are at the most only slight 
 differences of quantity. If enough tests are made carefully the usual 
 grading will emerge on the average. But just as in the tests for grading 
 without respect to quantity, the single tests here show frequent reversal 
 of the results that appear on the average. This, of course, only emphasises 
 the approximate equality of the lower grades of fusion. 
 
 Those who are not familiar with the notion of fusion must be careful 
 to avoid misapprehension. The results do not suggest that intervals 
 such as the fourth and the thirds are hardly distinguishable. Fusion 
 does not apply to the peculiar aspect of a pair of tones that we call 
 their interval; but to the whole mass of sound formed by the two 
 tones in so far as they merge into, or blend with, one another, or in so 
 far as they appear to be one to a person who does not recognise them 
 as one interval or another, and yet, of course, hears them as a whole. 
 Nor do these grades of fusion imply that the musician does not know 
 and hear the fourth as a greater consonance than the third, or the third 
 than the second. The musician is highly practised; he has learnt by 
 long experience, from tradition, and by harmonic usage to classify the 
 pairs of tones as distinct grades of consonances in spite of the slight 
 differences that may distinguish them, so that these differences perhaps 
 seem greater and more decisive to him than they really are. This 
 distinctness of grading is very much increased by the sharp differences 
 there are between pairs of tones as intervals, whereby they are at once 
 recognisable and distinguishable. For whatever is musically associated 
 with a certain pair of tones can be attached to their distinctive nature 
 as intervals and so can be unfailingly recalled; whereas if it had to be 
 recalled by their indistinctive nature as fusions alone, the slightness of 
 
 w. P. M.
 
 18 DEGREES AND THEORY [OH. 
 
 the difference between the lower grades of fusion would make recall 
 very uncertain and liable to confusion, so that in turn the want of 
 distinction amongst the lower grades would be thrown into prominence. 
 In observing fusion the musician has the special difficulty of abstracting 
 from his highly trained knowledge and from the intervallic aspect of 
 tonal pairs. The Greeks, of course, could easily distinguish thirds from 
 seconds as intervals, but it took them long to compare the members 
 of their class of discordant intervals with one another so carefully as 
 to see that the third stood very high in the class and had so much fusion 
 in it that they could place it next to the fourth and even class it in 
 a certain sense as a consonance. Advance in observation and analytic 
 abstraction seems to account for their change of classification more 
 easily than the hypothesis of development does. Improvements in the 
 method of tuning may also have been of influence. But there is, as we 
 shall see, no doubt that the differences between all the intervals with 
 which we are so familiar nowadays has only been fully displayed by 
 the functions they have acquired in our highly developed music. 
 
 The explanation of fusion in the first place follows closely the 
 suggestions given by the description of its highest grades. Fusion is 
 degree of resemblance to the unity or balance of a single tone. Two 
 tones that fuse, blend with one another so as to appear more or less 
 evenly intermingled, whether the hearer fails to recognise their difference 
 or whether his natural aptitude and his practice enable him to recognise 
 them at once. The fusion is not altered by the ability to recognise 
 the constituent tones in spite of their fusion. To the talented musician 
 who practically never fails to notice both its tones, an octave is still 
 a high grade blend. Nor does his ear contrive to isolate these two 
 tones from one another so that they shall appear to him to be as separate 
 as they are when successive. 
 
 Our previous study of tone showed that the whole series of tones 
 from highest to lowest probably consists of one total series of (hypo- 
 thetical) particles of sound, of which a number always beginning at 
 the same ('high') end of the series enters into any tone sufficient to 
 make up its volume. The highest audible tone requires approximately 
 only the single ('highest') terminal particle, the lowest the whole 
 series. Now when two tones are given together, the volume of the 
 lower must include the volume of the higher and apart from some 
 special marks the two will not be very easily distinguishable. The 
 coincidence of the two volumes would probably of itself make the upper
 
 ra] OF CONSONANCE AND DISSONANCE 19 
 
 part of the total volume more intense. And it is even conceivable that 
 the upper volume might thereby be recognisable as such, so that the 
 listener could say : in this total sound there is besides the total extent 
 of sound, a sound of a certain smaller extent. And it may be supposed 
 that these sounds would appear even and smooth (or specifically tonal) 
 in so far as other and irregular changes of intensity are absent in the 
 two volumes. 
 
 From this point onwards the attempt might be made to construe 
 the whole nature of tones and their combinations without the use of 
 pitch in the sense, above expounded, of a central point of prominence 
 in the tone's volume. I shall not argue the attempt out in detail. It 
 will suffice to acknowledge its logical possibility here. The use made 
 of the notion of pitch in the following pages will of itself exclude the 
 real possibility of doing without it. 
 
 Pitch, as above shown, probably occupies a central position in tone 
 and makes it the balanced symmetrical system it is. If this is so, the 
 overlapping of volumes will yield a special case when the higher volume 
 lies exactly between the pitch of the lower volume and their common 
 higher terminal (cf. fig. 7, p. 199, c and c 1 ; or fig. 8, p. 200). The whole 
 volume thus constituted will differ from the isolated volume of the 
 lower tone only in the extra intensity of the upper half and in the point 
 of predominance that is the pitch of the higher tone. In so far as this 
 pitch is detectable, the upper tone would be as precisely definable as in 
 isolation. And in so far as the lower end of the upper tone can be felt 
 to coincide exactly with the pitch of the lower tone, or at least to deviate 
 from it when it does not so coincide, this particular case of simultaneous 
 tones would be very precisely definable. And the whole sound would 
 approximate more nearly to the nature of a single pure tone than would 
 any other form of coincidence of volumes and pitches. These things 
 justify completely the identification of the octave fusion with this 
 special case of overlapping. 
 
 A second special case would be given when the two special defining 
 points of the higher volume lay equally far away on either side from 
 the predominating point of the lower tone (cf. fig. 7, p. 199, c and g). 
 This case would give a lesser approximation to the balance of the pure 
 tone than does the octave. For in it two new points or breaks of the 
 smooth continuity of the lower volume have been introduced by the 
 presence of the higher. And so the whole would be more easily dis- 
 tinguishable from the pure tone or from one tone than would the octave, 
 whether the two pitches were recognised in the whole or not. By the 
 
 22
 
 20 DEGREES AND THEORY [CH. 
 
 unpractised person the two pitches would be more easily recognisable 
 than they are in the octave; for the musical mind, there would be less 
 interpenetration of the two tones although he might not be able to 
 detect any difference in the ease with which the two tones were recognised ; 
 to both there would be less balance or smoothness in the whole sound 
 than in the case of the octave. We may therefore identify this case 
 with the second grade of fusion the fifth. 
 
 And we are confirmed in so doing by the well known difference 
 between the octave and fifth namely that octaves are simultaneously 
 compatible with one another, whereas fifths are not. If we add a double 
 octave to the first, the third pitch-point falls exactly into the upper 
 half of the middle tone without disturbing the relative balance of the 
 two lower tones ; but if we add a second fifth, it will not thus fit in with 
 the first two tones. On the contrary, it entirely spoils the balance and 
 symmetry of the first fifth. 
 
 It follows from the nature of the balance claimed for the fifth that 
 the volumes of its two tones are as 3 to 2. And along with the octave 
 case this implies that the volumes of all tones that have the ordinary 
 fusional relations to one another, are inversely proportional to the 
 corresponding rates of physical vibration. But the relations of the 
 volumes have been educed without any appeal to this familiar physical 
 fact, at least, logically. So long as this is so, it is a matter of impossible 
 speculation to discover whether, given the proper analytic approach 
 to the problem, the volumic ratios would have been discovered and 
 sufficiently proved, had the physical knowledge not preceded. Our only 
 concern need be whether sufficient ground now exists upon which to 
 raise a logically independent proof. This undertaking is not a piece of 
 mere pedantry, as some might think who recall that one way of proving 
 a thing is enough. For the proof given has not for its object the re- 
 proving of truths concerning physical ratios, but the demonstration of 
 an entirely new object, namely the ratios of the volumes of tones, 
 solely as they are heard. This object is a psychological one, if you like. 
 It deals with an object that is as different from physical vibrations as 
 blue is from commotions of the ether or as thought is from brain process. 
 
 It is as absurd to suppose that the only objects we can study 
 logically and scientifically and convincingly are physical as it would be 
 to suggest that we could not think correctly until we knew the physiology 
 of the brain perfectly. You may reply that of course we think correctly 
 because the brain is there working correctly whether we know of its 
 workings or not. True; and when we prove things properly about
 
 ra] OF CONSONANCE AND DISSONANCE 21 
 
 tones, the brain doubtless works properly too. But the chief interest 
 for us is to think correctly first. There is, of course, no doubt that the 
 brain must be capable of interacting in some special way with physical 
 sounds if we are to hear sounds and to think about them correctly. 
 But experience is more than brain. And men may yet have to infer 
 something from the study of sounds as they are for us when heard, if 
 they are to understand how the brain acts in connexion with them. 
 Besides, brain and mind or experience, or more specifically, brain and 
 tones or music are two very distinct and different things that no one 
 could possibly confuse with one another, no matter how much they 
 may be dependent upon, and may interact with one another. So each 
 of them must be studied for its own sake and in the special way its 
 peculiar nature and its relation to ourselves make possible. We cannot 
 bring our knowledge of both into harmony until we have attained a 
 complete knowledge of each. 
 
 The form of overlapping of the other intervals and the kind of 
 balance between the parts of the volume they constitute may easily 
 be reckoned out from the knowledge we have just gained. In the case 
 of the fourth, the higher volume will be three-quarters of the length 
 of the lower volume. So the lower limit of the upper tone will fall exactly 
 half-way between the lower limit of the lower tone and its pitch-point. 
 And the pitch-point of the upper tone being at its centre will lie three- 
 eighths of the length of the lower tone from its upper terminal and one- 
 eighth from the pitch-point of the lower tone. The whole mass of sound 
 of the two tones will be marked into parts of two, two, one, and three, 
 eighths of its length. There is here some balance in the one half of the 
 whole volume and less in the other. For the major third the parts 
 are 2, 3, 1, and 4 tenths, which seem more irregular. 
 
 As we proceed, the disproportion of the parts seems to grow greater 
 and one or other of the points of the higher tone falls nearer and nearer 
 to the pitch of the lower one. In so doing it must, of course, become 
 more noticeable; just as we catch sight of a second visual point the 
 more easily the nearer (within ordinary limits) it lies to another that 
 we notice easily and are attending to. We notice the pitch-point of 
 the lower tone more easily because it lies in the centre of the whole 
 volume of sound formed by the two tones; and as pitch is always 
 central in tone and we are constantly at work with the pitches of tones 
 in music, so we get into the habit of attending to tones in a central 
 balanced manner and then notice the pitch and volume of the lowest 
 component of a chord most easily. We do so, not only when the chord
 
 22 THEORY OF CONSONANCE [CH. HI 
 
 is an isolated stationary mass of sound, but also in music where each 
 tone is a phase of a voice or part, except in so far as some melodic 
 figure or theme is present that specially makes out another element 
 than the lowest one of the moment. 
 
 But a study of the proportions of the parts of the lower fusions 
 shows no great differences between them. That accords well with the 
 slight differences they show for the ear. And there seems to be no 
 ready way in which we might make our conceptual or theoretical 
 treatment of these volumic parts give us a much more exact account 
 of the degrees of fusion of intervals than the ear gives, so as to provide 
 a rule or rapid guide to the ear, as e.g. measurement is for many purposes 
 a guide and control to the eye. That deficiency does not indicate that 
 the volumic theory of fusion is wrong. The agreement the theory 
 shows between concept and sound, speaks only in its favour. Measure- 
 ment is by no means an infallible guide or a standard of visual art. 
 It is possible, however, that calculation may yet discover more delicate 
 or more adequate ways of representing the volumic basis of fusion, 
 and from following and moulding itself to portray the verdicts of hearing, 
 may gain strength to place itself ahead of the sense and to lead it for- 
 wards to unexplored regions where it may dwell with pleasure and 
 where its art may flourish more abundantly. If that is possible, it would 
 be foolish not to cherish the idea merely because theory has rarely, 
 if ever, yet preceded the experiments of musical art. On the contrary 
 we have every reason to hope that theory will yet be as great a support 
 to art as it has come to be to industry. And we shall do well to build 
 hopes for art upon this dream. After all, such theory does not wish to 
 import into art influences and notions entirely extraneous to its matter, 
 as are ratios of vibrations and all merely physical knowledge. At its 
 best such theory is really only a most perfect description of the actual 
 things that the art of music deals with, tones and their groupings. It 
 gives the artist a better comprehension of their real being than he 
 would gain from his ear alone. It is merely the hearing of the ear 
 perfected and purified by the wide attention and analysis of the intellect. 
 If the art of music can turn such work of the intellect to fruitful use, 
 why should it not be allowed to profit thereby? After all men have 
 had ears and eyes since the dawn of time; but they have not always 
 had the minds to make art out of sounds and sights. Their minds 
 have had to grow to the power of that creation.
 
 CHAPTER IV 
 
 THE RELATIONS OF FUSION TO BEATS, PARTIALS, AND 
 DIFFERENCE TONES 
 
 HAVING attained what seems to be a satisfactory explanation of fusion, 
 one that is grounded upon the fusing tones themselves and does not 
 involve any reference to any other' phenomena that may accompany 
 the simultaneous occurrence of two tones, we may feel free from any 
 obligation to refute those theories that base their explanation of 
 consonance solely upon such adventitious phenomena. Besides these 
 theories have already been well refuted (67); so that before a theory 
 of consonance by volumic balance had become attainable, the dis- 
 cussion of the problem had yielded the conclusion that the grades of 
 fusion are an immediate characteristic of pairs of simultaneous tones 
 and that the most probable explanation of these grades was to be 
 sought in some form of sensory 'synergy' (64, 214). An analogous case 
 is familiar in vision, where certain colours are found to look well together, 
 while others 'kill' their neighbours. We have no very satisfactory 
 explanation of these peculiar harmonies of vision ; but the most probable 
 theory supposes that the physiological processes underlying in- 
 harmonious or discordant colours in the retina or in the central areas 
 of the brain that subserve vision, do not work easily together or do 
 not make material for each other, as it were, or predispose each other's 
 functions. The notion of 'synergy' is, then, not a specific theory so 
 much as an attempt to indicate where the basis of a true theory will 
 probably be found. And the volumic theory given above may quite 
 well be looked upon as a solution of this query as to the exact nature 
 of fusional 'synergy.' 
 
 This by no means recent (1893) reduction of the field in which the 
 explanation of fusion may be sought seems to be still unknown to many 
 of those who are interested in the theoretical foundations of music 1 . 
 
 1 Thus, for example, in the year 1917 Shirlaw (60, *si) writes: "The only thing which 
 theorists who have made the harmonic series the principle of chord generation appear 
 to have omitted to do has been to abide by the results of their own theory. Having 
 accepted a fundamental and guiding principle of harmony, they have nevertheless refused 
 to be guided by it, and have virtually abandoned it, or, while still professing to do it 
 homage have vainly attempted to exploit it for their own purposes. The principle of 
 harmony of Zarlino, Descartes, Rameau, Tartini, furnishes us with but a single chord.
 
 24 THE RELATIONS OF FUSION [OH. 
 
 It may therefore be profitable to make a short review of the critical 
 part of this work now. In so doing we shall best follow the exposition 
 given by the author of the theory of 'synergy.' We shall thereby not 
 only do justice to an important phase of the development of the science 
 of sound, but we may also reach a greater completeness of scientific 
 outlook. For it is quite possible that once the volumic basis of fusion 
 is given, its nature may be heightened or lessened by these other 
 adjuncts of the fusing intervals, even although they are incapable of 
 producing by themselves the effect of consonance or dissonance. 
 
 The most familiar and the most widely accepted theory of consonance 
 is that of Helmholtz. Helmholtz gives, as Stumpf (67, 2) pointed out, 
 "not one, but two different definitions of consonance and dissonance, 
 which are indeed closely interwoven in his exposition, but which really 
 are quite different and apply to different fields." According to Heffernan 1 
 (19) there are indeed no less than three elements in Helmholtz's explana- 
 tions of consonance. 
 
 The chief definition is: "consonance is a continuous, dissonance an 
 intermittent sensation of tone" (20,226). The intermittence is here 
 created by the beats which appear when tones of about the same number 
 of vibrations per second occur together. As far as they are audible, 
 the number of beats is equal to the difference between the numbers 
 of the two rates of vibration. These beats may originate from the 
 primary tones (as in the interval of a minor second) or from the upper 
 partials of the primaries (as when the fifth partial e 3 of c 1 beats with 
 the fourth partial f 3 of f 1 ) or from a partial of the one tone with the 
 other primary (as when c 2 beats with the second partial of 6); it is 
 a matter of indifference what their source is, so long as they make 
 the binary sound noticeably rough. For they must then certainly help 
 to make dissonances less suitable for pleasure and for the clear exposition 
 
 But this ought not to be regarded as a negative result, but as a positive result of the 
 greatest theoretical significance. It is the one fact of supreme importance which this 
 principle has to teach us. This has not yet been realised.... There exists in our harmonic 
 music but a single chord, from which all others are developed. But as the sounds of this 
 harmony are contained in the resonance of musical sound itself, all harmony has its source 
 in a single musical sound. The development of harmony has been a more simple and 
 beautiful process than musicians and theorists have imagined." And in a footnote to the 
 word "developed" Shirlaw promises "a new and smaller constructive work on the theory 
 of harmony." 
 
 1 This paper (1887) gives a striking criticism of Helmholtz, but its experimental 
 basis is at times insufficient and lacking in precision.
 
 iv] TO BEATS, PARTIALS, AND DIFFERENCE TONES 25 
 
 that art requires than they would otherwise be. And if beatless sounds 
 are of themselves smooth, contrast with the roughness of beating 
 chords will make them seem smoother and more beautiful still. 
 
 But it is important to notice that Helmholtz's theory can by no 
 means justify the ascription of smoothness to beatless chords. Smooth- 
 ness is not a mere negation; a table is not smooth to touch as soon 
 as it ceases to be rough; it is smooth only so long as it gives the finger 
 a continuous area of unvaried sensation. And the regularity and positive 
 smoothness we have shown to exist in tone make any such purely 
 negative use of smoothness inadmissible. Still it is evident that the 
 roughness of beats will heighten and increase the general effect of the 
 asymmetry and unbalanced disproportion of parts that we have shown 
 to constitute the primary being of dissonance. It will make differences 
 and grades of unsuitability and unpleasantness amongst the lower 
 grades of fusion that might otherwise be less distinctive. 
 
 So far the theory would thus account only for grades of 
 unpleasantness amongst combinations of simultaneous tones. For 
 successive tones the explanation fails altogether, since no beats then 
 appear in any case. Here Helmholtz appeals to his second basis of 
 consonance in the partials that are common to the successive tones. 
 In the octave we hear again the largest possible part of what we heard 
 before in the lower tone, namely every even numbered partial. In 
 the fifth we hear a lesser part, but still much, namely every partial 
 that is a multiple of three. And so on (20, 253 ff.). We do not need to 
 have analysed these partials from the whole tone by special attention 
 and to know of them. The second tone merely appears to be like the 
 first one according to the amount of identity amongst the partials of 
 the two, as two faces often look similar without our being able to say 
 in what respect precisely they are so. 
 
 This second theory is in a notable respect the obverse of the first, 
 as it were. For now the positive quality belongs to consonance alone. 
 Dissonance is a mere negation of consonance; two tones are dissonant 
 when they have nothing in common ; or dissonance is the lowest degree 
 of consonance and yet it is not consonance at all. Moreover it is clear 
 that the continuity thus established between successive tones would 
 be a real influence connecting them, in so far as it could be felt in the 
 aggregate: and though it would necessarily of itself be only a weak 
 bond between tones, it would undoubtedly come to the good of any 
 obvious bond already linking consonant tones to one another. 
 
 But Helmholtz's second theory does not apply to simultaneous
 
 26 THE RELATIONS OF FUSION [CH. 
 
 tones. For the partials that are common to two tones an octave apart 
 could at the most only constitute a set of further primary tones to the 
 two from which they originate. Of course they do not under ordinary 
 circumstances do anything of the sort. Besides, Helmholtz's theory 
 is generally quite unable to explain satisfactorily why a tone and its 
 partials are heard as a unit and not as a number of tone spots or separate 
 tones. 
 
 "We have thus in fact two different principles in Helmholtz's theory, 
 the one valid exclusively for simultaneous, the other exclusively for 
 successive tones. This state of things seems strangely to have escaped 
 his notice, and as he himself nowhere emphasised the twofold nature 
 of his definition, it has also generally and from the first not been felt 
 to be a defect" (67, 4; cf. 34, 160). Such a duplication and crossing 
 of explanatory principles is certainly derogatory to any theory of the 
 system of consonances and kindred relations in successive and simul- 
 taneous tonal groups. That system nowhere suggests any such dual 
 nature; and we could hardly expect two such different causes to yield 
 so homogeneous a system of phenomena. In face of the positive theory 
 of consonance given above these logical and phenomenal inconsistencies 
 would be enough to refute Helmholtz's theories alone. But the following 
 arguments have been stated besides (67, 4ff.). 
 
 As beating is a periodical change of tonal intensity, such inter- 
 mittence of sensation can easily be produced either in a single tone, 
 or in each of two simultaneous tones. A sounding instrument may, 
 for example, be placed in a closed box from which a tube leads to the 
 surface of a rotating 'disc; in this disc as many holes as need be are 
 pierced, so that they pass the mouth of the tube when the disc is rotated. 
 Dissonance does not then appear any more than it does when a consonant 
 interval is played as a tremolo. Besides such beats without dissonance, 
 we can have dissonance without beats, as from tuning-forks on their 
 resonance boxes sounding to 500 or 490 and to 700 vibrations, or to 
 700 and 1000, or to 780 and 1100. Such forks well sounded contain 
 hardly more than a trace of the first partial and at these rates of vibration 
 beats are quite inaudible. Moreover when forks without their resonators, 
 preferably between 800 and 1200 vibrations, are held one before each 
 ear, their tones are not carried to the opposite ears, either by the air 
 or by the bones of the head, so long as they are not made too loud. 
 When two forks of say 800 and 900 (a major tone) are tested in this 
 way, only a trace of beating can be heard at the very most; in striking 
 contrast to the effect of both forks before a single ear; but the dissonance
 
 iv] TO BEATS, PARTIALS, AND DIFFERENCE TONES 27 
 
 in the two cases remains the same. When in turn a consonance, e.g. a 
 major third (620 and 775 vibrations), is tested thus, it remains just as 
 fused as usual. 
 
 The number of beats accompanying any dissonant interval (e.g. a 
 major second) must gradually increase as the interval is raised through 
 the musical range of pitch; but there is no indication that it therefore 
 becomes a dissonance only at a certain pitch or ceases to be a dissonance 
 in the higher octaves. Nevertheless the roughness that is due to beating 
 and that accompanies dissonance under ordinary circumstances will 
 indeed vary with the audibility of the beating (which varies with its 
 rate). Nor do differences of timbre make such regular differences in 
 the degree of dissonance or in the musical usage of intervals as they 
 might be expected to do in view of the different possibilities of beating 
 they create. 
 
 But it is needless to follow the argument further. Let us consider 
 the alternative. 
 
 The chief argument against consonance by coincidence of partials 
 is its continued appearance amongst tones devoid of partials. And, 
 Helmholtz's work on timbre shows that the different musical instru- 
 ments, far from having each the complete series of possible partials, 
 differ precisely in the selection from the series that is typical of them; 
 and yet the grades of consonance do not differ from instrument to 
 instrument or from one intonation to another. Suppose, for instance, 
 that the clarinet has, as Helmholtz says 1 , only the uneven partials; 
 then an octave on clarinets could not possibly be a consonance at all, 
 but rather an extreme dissonance, because in the second tone we should 
 hear nothing of what we heard in the first. By means of 'interference' 
 we can exclude from a tone any specified partials so as to make coinci- 
 dence unattainable. Consonance, however, remains unaffected. The 
 consonance of tuning-fork tones is beautiful in spite of the restriction 
 of their partials. Helmholtz may well have been aware of the main- 
 tenance of consonance in spite of the relative purity of fork tones. 
 However he may have adjusted his mind to this fact, his successors 
 at least have variously appealed to the influence of memory. But, 
 as Stumpf says (67,16): "The remembrance that two other blends 
 on the same fundamentals once were consonant, can only bring the 
 
 1 According to D. C. Miller's harmonic analyses the seventh, eighth, ninth and tenth 
 partials predominate in the blend of the clarinet. "The seventh partial contains eight 
 per cent, of the total loudness, while the eighth, ninth and tenth contain 18, 15 and 18 
 per cent, respectively" (40, *>i). The second, fourth and sixth partials are very weak.
 
 28 THE RELATIONS OF FUSION [CH. 
 
 non-consonance of the present tones by contrast more strongly to my 
 notice. A dish that lacks salt would never be said to be well salted by 
 mere force of memory or custom; on the contrary." 
 
 " In short, timbre is for one and the same interval extremely variable, 
 but the degree of consonance is constant. Hence both cannot be 
 explained from one and the same principle. And it is just the happy 
 explanation of timbre that Helmholtz achieved for acoustics for all 
 time that makes his explanation of consonance from the same principle 
 an impossibility" (67, 19). 
 
 Various attempts have been made to bring consonance and its 
 grades into relation to another class of phenomena that accompany 
 the simultaneous occurrence of two or more tones, namely difference 
 tones. The earliest such attempt was made by their discoverer x Tartini 
 after whom they were often called ' Tartinian tones.' Helmholtz appealed 
 to them in explanation of the less harmonious effect of the minor, as 
 compared with the major, triad, thus introducing a third factor in the 
 creation of consonance. Of recent years an elaborate attempt has been 
 made to base the grades of consonance solely upon the beating and 
 confusion (of a special kind) of neighbouring difference- tones. On the 
 basis of numerous observations of the difference-tones that accompany 
 intervals of different ratios, it was claimed that the greater dissonance 
 had the greater number of difference-tones within close pitch-distance 
 of one another, and would therefore have the more beating and confused 
 blurring. That is to say the latter constitute dissonance 2 . 
 
 This type of theory seems to escape the criticism fatal to Helmholtz's 
 explanation by coincidence of partials, that its basis is withdrawn 
 when the primary tones are purified of all partials. For difference-tones 
 still accompany such pure tones. They are not due, like partial tones, 
 to any physical process in the sonorous body or in the air between that 
 and the ear, but they arise somewhere within the ear directly from the 
 
 1 Or 'one of their discoverers.' Cf. 60, soi: "Although Tartini is generally regarded 
 as the first to discover the combination tones he had asserted that as early as 1717 he 
 had made use of them for the purpose of teaching pure intonation on the violin to his 
 pupils it is certain that other musicians had discovered them independently. J. A. Serre 
 of Geneva, and Romieu of Montpellier, had given accounts of these tones before Tartini' s 
 publication of the Trattato di musica (1754)." G. A. Sorge in his Vorgemach der musi- 
 kalischen Komposition, "published nine years before Tartini's Trattato di musica, demon- 
 strates his acquaintance with the phenomenon of combination tones." 
 
 2 For sources and criticism see 68, 57. For the most trustworthy and complete record 
 of observations of difference-tones see 70.
 
 iv] TO BEATS, PAKTIALS, AND DIFFERENCE TONES 29 
 
 interaction of the primaries. But difference-tones can be greatly 
 weakened, if not made to disappear entirely when the primary tones 
 are presented one to each ear and are given in somewhat weak strength 
 (cf. 7). The dissonance, however, does not then disappear nor change 
 its degree. Besides it is a fatal defect of this type of theory that it gives 
 no really positive status and explanation- to consonance. Consonance 
 is here a mere negation or minimum of dissonance. And whatever be 
 the nature and cause of dissonance that is postulated whether it 
 consist in the multiplicity of the difference-tones or in the fluctuations 
 of their beating or whether it be traceable rather to the confused 
 indistinguishability of too closely neighbouring difference-tones that 
 form between-tones or even a sort of streak-tone, or the like we 
 should in any case certainly have no reason to hear non-dissonant 
 sets of tones in any other way than with the greatest precision and 
 clearness of distinction from one another (cf. 68, 282). Even though, 
 as in the great consonances, the number of difference-tones is greatly 
 reduced (to none in the octave), the two primary tones would still be 
 two in all obviousness; there could be no excuse for holding them to 
 be but one, and no ground for establishing any special relation between 
 them (such as that of 'consonance') except that of clear distinguish- 
 ability. Thus the appeal to difference- tones can only give a partial 
 explanation and must therefore be unsatisfactory. 
 
 If, however, a positive explanation of consonance and dissonance 
 in their grades has already been given, as in the previous pages, it is 
 obvious that any beating of difference-tones amongst one another or 
 with the other components of the whole sound would add to the rough- 
 ness and irregularity inherent in the latter through its primary com- 
 ponents, while the consonance of these parts which would have to 
 rest upon the same kind of processes as the consonance of the primaries, 
 would bear out the latter. Consider, for example, the case of the 
 perfect fifth in pure tones in relation to the two loudest difference- 
 tones the 'first' (h-l) and the 'second' (2l-h). The ratio for the 
 fifth is 2 : 3. Its difference- tones are both of ratio 1. A slight mistuning 
 will yield one difference- tone just less than 1 and another just more 
 than 1. These two will beat with one another, whereas in the just 
 interval we shall have three consonant intervals, octave, fifth and 
 twelfth. Similarly in the two common triads, major and minor, whose 
 ratios are 4:5:6 and 10 : 12 : 15 respectively, we find the following 
 components : in the major chord, 1 (twice), 2 (twice), 3, 4, 4, 5, 6, or 
 C, c, g, c 1 , e\g l ; in the minor chord, 2, 3, 5 (twice), 8, 9, 10, 12, 15, or
 
 30 PARTIALS AND DIFFERENCE TONES [CH. iv 
 
 A\>, E\>, c, a\>, &[?, c 1 , e 1 ^, g 1 . Apart from the difference of octaves 
 there are three dissonant intervals in the latter chord and fewer high 
 grade consonances (3 octaves, 6 fifths, and 2 fourths to 4 octaves, 
 5 fifths, and 1 fourth). We shall return to this topic again (p. 192 f.). 
 The general series of the upper partials and the difference-tones 
 were on the whole a relatively late discovery in the history of the 
 scientific foundations of music. But many years before Helmholtz 
 propounded his very convincing theory of instrumental tone-blend 
 (timbre), they had become familiar to all the leading theorists. If any 
 feature at all of tones were really explicable in terms of some such 
 adventitious accompaniments of primary tones, or rather consisted of 
 them, we might certainly have expected Helmholtz's predecessors to 
 have learned how to explain pitch-blend by the grouping of partials. 
 That they did not do so, and obviously were not tempted to do so, 
 gives us the right to consider it highly improbable, apart from all other 
 grounds, that so direct and unmistakable phenomena as those of 
 consonance and dissonance are founded upon such remote accompani- 
 ments of primaries as partials and difference-tones and their beatings. 
 We must find the basis of consonance and dissonance, as it were, 
 directly in or below the primaries themselves. And that the theory 
 propounded above has succeeded in doing.
 
 CHAPTER V 
 
 THE CONSONANCE OF SUCCESSIVE TONES 
 
 WE have not yet given an account of the consonance of successive 
 tones from the standpoint of the volumic theory. But it is evident 
 that the task is a very simple one and involves no change of the basis 
 of explanation and no new principle. This necessity characterises all 
 the other theories we have noticed. There is no beating between 
 successive tones, so that there can be no roughness between them. 
 And while one tone may certainly repeat a number of the partials of 
 a preceding one, yet there is no means of detecting which of the partials 
 appearing with two simultaneous primaries belong to either, in so far 
 at least as they might belong to both. Finally, the difference-tones 
 that appear with simultaneous tones are lacking in their sequence. 
 
 The attempt has been made to cover over these lapses of the basis 
 of explanation by appeal to the restorative work of memory! The idea 
 is that when the basis of consonance or dissonance is actually given, 
 the memory will mark it well and associate it with the primary tones 
 which it accompanies. Later when these primaries appear without 
 .the basis of their fusion, this characteristic will be restored by the 
 memory and the primaries will function as fused. There is no general 
 psychological fault in this theory so far as the memory's activity is 
 concerned. Seeing a man often and hearing him speak, we learn to 
 connect his voice with his visual appearance; when we later merely 
 see him we can call his voice vividly to mind.' 
 
 But we do not then hear him speak. Memory does not cause hallu- 
 cinations in the most of us, nor should we desire it to do so. There 
 are, however, cases of a less abnormal character which seem to imply 
 a restoration of sensation by memory. Thus a glowing iron is often 
 said to look hot, a child's cheek looks soft and tender, the ground after 
 rain looks wet. True; but these things do not then feel hot or tender 
 or wet; they merely look so, because their visual appearance makes us 
 think at once of the associated character that comes through the other 
 sense. There is a certain visual feature in each which prompts the 
 mind to recall the associate, and so that visual feature acquires in our 
 minds a special meaning as a sign of the associate. But the visual 
 'ground' no more feels wet because it looks so, than the word 'lead'
 
 32 THE CONSONANCE OF SUCCESSIVE TONES [CH. 
 
 acquires a weight because it is the sign of a heavy thing. On the con- 
 trary when a thing looks heavy, it usually feels lighter than another 
 thing of the same weight that does not look heavy or that looks light. 
 This is the size- weight illusion, demonstrable with two objects of 
 equal weight but of different size. Expecting weight we do not feel it 
 to be greater than usual, but less. 
 
 At least as much as this is also true of tones in relation to consonance. 
 If you hear c and then d and recognise the interval between them 
 specifically as a whole tone or merely as much smaller than the consonant 
 intervals, you may certainly recall the fact that these tones together 
 would form a dissonance. But they would not therefore sound dissonant 
 in sequence. On the contrary, if you had learnt from the simultaneous 
 tones to expect a dissonance, then on hearing them in sequence you 
 would be greatly struck by the absence of dissonance: just as you would 
 be astonished by the weight of a cigarette you had picked from a box 
 if it happened to be made of lead. 
 
 The suggested explanation of fusional degrees by 'synergy' has 
 met with a similar difficulty in explaining the relations of successive 
 tones to consonance. A solution by presumption offers itself readily 
 enough, however. For we may suppose that the special function by 
 which two tones make each other's action easier or harder when 
 simultaneous, still exists when they are successive. For the earlier 
 tone is not then entirely gone, any more than the earlier part of a melodic 
 phrase is mentally non-existent when the later notes are being played. 
 The function of fusion would then hold between tones that are ' together 
 in the mind,' so to speak, whether simultaneously or successively. In 
 other words the mind's sphere of immediate activity, unaided by 
 memory, covers not only the present instant 'now,' but a short reach 
 or length of time. Of course, we should still have to explain why in 
 sequence tones do not fuse in the same way as when simultaneous. 
 
 This way of accounting for the relations of consonance both to 
 simultaneity and succession of tones by the same principles seems 
 easy only because no definite theory has been advanced. Only the 
 formal requirements of a successful explanation have been sketched. 
 The other theories, such as Helmholtz's, failed decisively because they 
 claimed to have found a definite cause of consonance or dissonance 
 which criticism has shown to have apparent validity only in respect 
 of simultaneous or successive consonance and to be obviously inapplicable 
 to the complementary case. And Stumpf did not feel quite satisfied 
 with this extension of his notion of ' synergy ' ; for he suggested various
 
 v] THE CONSONANCE OF SUCCESSIVE TONES 33 
 
 
 
 means whereby the relations of successive tones might be brought 
 into closer parallel with the fusion of simultaneous ones, including the 
 principle of relationship through partials, advocated by Helmholtz 
 (cf. 19, 58 ff.). Coincident partials may well give another kind of bond 
 between tones, but that cannot be the bond of consonance, if consonance 
 is to be explained by 'synergy.' We need not debate these notions 
 further now. Let us rather consider the problem from the volumic 
 point of view. 
 
 It is immediately clear that two tones an octave apart in sequence 
 must have a special relation to one another as volumes. The higher 
 one will fall in the tonal field exactly upon the upper half of the volume 
 that formed the lower tone. Or the lower will occupy just twice the 
 volume occupied by the previous higher tone; its pitch-point will be 
 exactly where the lower end of the volume of the higher tone lay. 
 When the tones are simultaneous, we notice how perfectly the two fit 
 together to form a regular whole. Perhaps we get our impression here 
 more as a whole than from an analytic study of the coincidence of points. 
 The coincidence is there, of course; but we probably feel the fit as 
 a whole rather than see it or inspect it point for point. When tones 
 follow one another, however, this analytic procedure becomes more 
 possible. We could not, of course, state in exact conceptual terms our 
 procedure in observing the tones, so as to corroborate precisely the 
 theory of their volumes. But the different ways we use our attention 
 might really correspond to the statements we deduce from the theory 
 of the volumes of tones for all that. 
 
 Thus, e.g. in noticing the lower tone after a few trials we may well 
 fixate the pitch of it exactly and observe then whether the lower end 
 of the higher tone just touches off that pitch-point. In vision we can 
 describe the procedure of the attention in very precise conceptual 
 terms. We take one line and let the end point of it fall exactly on the 
 end point of another line and make a second point of the line fall on 
 some other point and so on. We are unable to do this in hearing, not 
 because the stuff of sound would not allow of it, nor because our minds 
 are somehow befogged in dealing with tone, but simply because we 
 cannot turn and move tones about in the auditory field as we move 
 figures in the visual field. Nor can we dot any required point into a 
 sound volume as we do with visual lines, and so on. 
 
 Similarly in the fifth we pass from the one tone to the other by an 
 easy path. We could not fail to notice the symmetrical relation of the 
 
 w. F. M. 3
 
 34 THE CONSONANCE OF SUCCESSIVE TONES [OH. 
 
 new defining points of the higher volume to the pitch of the lower, 
 even if it were quite impossible to sound two tones at once. 
 
 But it is obvious that there is a considerable difference between 
 simultaneity and succession. The former creates a balanced, unitary 
 mass that differs from other such masses in its degree of balance and 
 unity. Sequence creates a passage that may be regarded in much the 
 same way. The lower tone gives way to its octave gracefully, as it 
 were; it almost introduces it, pointing in a sense to the place where 
 it will appear, or preparing a place for it against its coming. The same 
 is true in a different manner for the fifth. 
 
 At the same time it is quite possible for the mind's eye to take the 
 measure of the two tones in volumic projection upon one another, as 
 it were, and to see their volumes against one another as if they were 
 simultaneous, without, of course, being so. That is, we can, if we will, 
 take a sequence of tones as if they were a fusion of simultaneous tones 
 and judge them accordingly. The two tones do not, of course, actually 
 fuse ; but they have to be taken or heard together as if the first one were 
 still there when the second appears ; their intensities are not summated 
 as in the case of simultaneous fusion (cf. p. 52 f. below), but the balance 
 and symmetry of their relative positions are noted. This is done 
 regularly in music in the arpeggio forms of chords. But it is not 
 necessary on the other hand that the mind should always do so. There 
 is not only no reason why it should, but it is easy for it to do otherwise. 
 
 Special interests of music, especially . the melodic, lead us to take 
 successive tones specifically as a sequence. Here, on the contrary, 
 the tones are apprehended specially as a sequence; we let the first 
 one go and pass from it to the second. In the arpeggio chord we have 
 a whole given successively ; in the melody a sequence or motion is given 
 successively. Or in the chord the successive tones are held in projection 
 upon one another, while in the melody they are each complete stages 
 of a transition. For this purpose smallness of interval is a favouring 
 factor; it makes for continuity or for melodic progression 1 . Continuity 
 is present even with the larger interval, but it is then not so obvious 
 or so obtrusive; it may have to be supported by other relations which 
 bring successive tones into connexion with one another. In this way 
 
 1 Cf. 41, 246: "In folk music generally the frequency with which the various intervals 
 are used decreases proportionately with their size." It does so also in the melodies of 
 Schubert's songs, as I have determined by sampling every tenth song. Only the minor 
 second occurs less frequently than the major. The figures of the several frequencies are : 
 21673, 112171, 3926, III-455, 4633, T 60, 5195, 6118, VI 58, 717, 
 VII 0, 028, 91, IX 0, 101, X 2. The sample consisted of 56 songs.
 
 v] THE CONSONANCE OF SUCCESSIVE TONES 35 
 
 we very often find the melodic and the consonantal aspects conjoined in 
 the same interval. Large intervals enter into melodies more easily when 
 they are such as would be consonances with simultaneous tones (cf. 
 52, 22) 1 . Not that the greatest consonance the octave is oftenest 
 used, the fifth next, and so on. Each interval has to be judged on its 
 merits. The great consonance of the octave is weighed down by the 
 large melodic step required by it and is probably less often used for 
 that reason. The fifth with a lesser consonance will very likely be used 
 oftener because of the greater advantage given by its much smaller 
 step. The matter has not been fully treated statistically, as far as I 
 am aware, but it would probably well repay the trouble necessary to 
 gather the facts. 
 
 Thus it seems that in the volumic theory a basis is presented from 
 which all the interests of music in simultaneous and successive tones 
 may be fully satisfied without neglect of any of the differences involved 
 in these two cases. 
 
 1 This is confirmed in the statistics of Schubert's songs. 
 
 32
 
 CHAPTER VI 
 
 THE NATURE OF INTERVAL 
 
 THE nature of interval has always been one of the great mysteries of 
 sound. It formed for the ancient world a fitting parallel in sense to 
 the wonderful relations shown by numbers. The discovery of the 
 connexion between the grades of consonance and the ratios of the 
 smaller numbers let loose a flood of mysticism which endured for 
 centuries. Rameau seems even to have thought that a thorough 
 explanation of the sensory basis of tonal proportion might lead to an 
 insight into the being of proportion in general and in particular as it 
 appears to us in numbers 1 . He was sharply criticised for this by the 
 Academic des Sciences to whom he presented his scientific plans for 
 approval and support. And their censure of his mystical vanities was 
 re-voiced by D'Alembert 2 in spite of the admiration which Rameau's 
 efforts and success in forming a systematic whole out of the empirical 
 musical wisdom of his time aroused in him. Rameau, of course, did 
 not succeed in explaining the mystery of interval and its relation to 
 
 1 53, 2: "Ne connoissant point la nature de notre Ame, nous ne pouvons appretier 
 les rapports qui se trouvent entre les differens sentimens dont nous sommes affectes: 
 cependant lorsqu'il s'agit des Sons, nous supposons qu'ils ont entr'eux les memes rapports 
 qu'ont entr'elles les causes qui les produisent." "Ce qu'on a dit des Corps sonores doit 
 s'entendre egalement des Fibres qui tapissent le fond de la Conque de POreille; ces Fibres 
 sont autant de corps sonores auxquels 1'Air transmet ses vibrations, et d'ou le sentiment 
 des Sons et de I'Harmonie est porte jusqu'a 1'Ame" (p. 7). "On peut dire meme quo la 
 Musique a cet avantage singulier. qu'elle peut toujours offrir en memo-terns a 1' esprit et 
 aux sens tous les rapports possibles par le molen d'un Corps sonore mis en mouvement; 
 au lieu que dans les autres parties des Mathematiques 1'esprit n'est pas ordinairement aide 
 par les sens pour appercevoir ces rapports" (Epitre). 
 
 2 "Le corps sonore ne nous donne et ne peut nous donner par lui-meme aucune idee 
 des proportions. . . .3. (et c'est ici la raison principale) parce que, quand on entendrait ces 
 octaves et ces sons des multiples, le sens de 1'ouie ne peut en aucune maniere nous donner 
 la notion de rapport et de proportion, que nous ne pouvons acquerir que par la vue, et 
 par le toucher. Pour avoir une idee nette des proportions et des rapports, il est necessaire 
 de comparer les corps par ces deux derniers sens; la perception des sons n'y contribue 
 absolument en rien, n'y ajoute rien, y est totalement ^trangere. Pour tout dire en un 
 mot, quand les homines seraient sourds, il n'y en aurait pas moins pour eux, des rapports, 
 des proportions, une geometric. En voila, Monsieur [Rameau], plus qu'il n'en faut sur 
 ce sujet; et les Mathematiciens trouveront a coup sur que j'en ai encore trop dit" (9, 2i3f.).
 
 CH. vi] THE NATURE OF INTERVAL 37 
 
 physical ratios. But he was certainly right in feeling that there was 
 something in the sensory experience to be explained which would tell 
 us how we feel proportion in one instance at least, whether that case 
 can throw any light upon other forms of mentally grasped proportion 
 or not. He did well to linger longingly upon the wondrous problem. 
 And D'Alembert's denial was somewhat too sweeping, at least so far 
 as the presence of proportion in hearing is concerned. 
 
 That we detect very special and precise features in our tonal 
 experiences correlated to certain very definite proportions in their 
 physical stimulus, should prevent a cautious, logical mind from asserting 
 point blank that proportion has absolutely no place in hearing. 
 
 D'Alembert, like so many others since his day, was convinced that 
 the 'metaphysics' or psychology of hearing would "according to all 
 appearances always remain covered with clouds." And yet he somehow 
 convinced himself at the instigation of Rameau's researches that the 
 principal laws of harmony could be deduced from a single experiment 
 (9, xxvii). But we now know that that idea is really as absurd as 
 Rameau's speculations on proportion seemed to D'Alembert himself. 
 In fact it is worse. For Rameau did include the phenomena of hearing 
 in his field of search, whereas D'Alembert seems to have thought that 
 a physical experiment or relation was worthy of the place of honour 
 at the feast of music without wearing the garment of experience. 
 Criticism has since thrown that and all other intruders out into the 
 limbo they belong to. And the clouds have blown away. 
 
 In fact the solution is not by any means difficult to attain or to 
 apprehend, once the fundamental secret of tone has been discovered 
 and understood. However that may be, the wonder of it all remains 
 that sound and hearing should be so cunningly devised; that the weft 
 of nature's mighty looms should reveal so beautiful a pattern in this 
 auditory part. And the greater wonder still is that our intellect should 
 have been able from this slender basis to raise the great art of music 
 to such complexity. Our task in the following pages will be to try to 
 show how the great edifice of music is placed secure on its foundations 
 and how it is carried upwards towards the art as we know it. No one 
 who follows this science of tone from its beginnings can fail to be struck 
 by the extraordinary nature of sound and the marvellous skill with 
 which music has been created by man. 
 
 The oldest theory from which an explanation of interval was sought 
 started from the obvious fact of vibration in the sonorous body and the
 
 38 THE NATURE OF INTERVAL [CH. 
 
 relation between pitch and rate of vibration. "This doctrine, first 
 taught by the illustrious founder of the sect [Pythagoras], adopted 
 and developed by Lasos, by Aristotle, by Euclid, and later by the 
 neo-platonists, has been formulated by Nicomachus, whose words 
 Boethius transmits to us. 'It is not,' he says, 'a single vibration 
 that produces a uniform sound; but the string, once set in motion, 
 gives birth to numerous sounds, because it impresses frequent vibrations 
 upon the air. But, as the rapidity of these shocks [of the air] is so great 
 that one sound is confounded in some way with the other, we do not 
 perceive the distance [that separates them], and it is as it were a single 
 sound that reaches our ears. Now when the vibrations of the low notes 
 and the high 'notes are commensurable amongst themselves (as for 
 example in the proportions indicated above), there is no doubt that 
 these common measures blend together and produce the unity of sounds 
 we call consonance'" (14, 96f.; 3, I, 31) 1 . Keeping in touch with the 
 progress of the physical science of sound, this doctrine has been carried 
 down to our own time. It was the basis of Euler's Tentamen novae 
 theoriae musicae ex certissimis harmoniae principiis, from which the 
 Table opposite his page 36 has been copied so often. Thus the octave 
 gives a pattern of this kind : : : : the upper line of dots 
 representing the waves of the higher tone, the lower the slower waves 
 of the lower tone. The pattern for the fifth would be thus : V : V : 
 (2 : 3). Probably the best and at the same time the most self-critical 
 statement this theory has ever received was made by a Scotsman, John 
 Holden, in an Essay towards a rational system of music, published in 
 Glasgow in 1770. The psychological analogies he brought forward are 
 admirable. Even in recent years the theory has been renovated by 
 Th. Lipps, who believed that these waves were carried to the brain 
 and transferred to subconsciousness, there being a unit of process in 
 the latter for each physical wave. Somehow this sequence took on for, 
 or in, consciousness the form of a smooth unity. The rhythmic coin- 
 cidence of the processes of subconsciousness that went on when two 
 tones were sounded, was supposed to be felt by consciousness as con- 
 sonance, want of rhythm or its puzzling complexity as dissonance. 
 
 1 "Non, inquit, unus tantum pulsus est, qui simplicem modum vocis emittat, sed 
 semel percussus nervus saepius aerem pellens multas efficit voces. Sed quia ea velocitas 
 est percussionis ut sonus sonum quodammodo comprehendat distantia non sentitur, et 
 quasi una vox auribus venit. Si igitur percussiones gravium sonorum commensurabiles 
 sint percussionibus acutorum sonorum, ut in his proportionibus quas supra retulimus, 
 non est dubium quin ipsa commensuratio sibimet misceatur, unamque vocum efficiat 
 consonantiam . ' '
 
 vi] THE NATURE OF INTERVAL 39 
 
 "It is not," as John Curwen said (8, 10), "that the mind actually counts 
 the relative number of vibrations and consciously ascertains that one 
 tone gives exactly half as many as the other. But by one of those 
 rapid though complex mental processes which are the marvel of the 
 philosopher, it feels the result" adding afterwards in a similar context, 
 "in a way the Great Creator only knows." 
 
 The earliest forms of this theory were a legitimate attempt to bring 
 into connexion the two ends of the psycho-physical process, where 
 knowledge offers itself most readily, the physical and the auditory. 
 But for later theorists, who realise the gap there is for all systematic 
 possibilities between merely felt grades of consonance and ratios of 
 physical vibration, whether of the air or of the ear, the theory is merely 
 an effort to make bricks of straw. Besides, as Rousseau noticed (56, 
 Art. 'Consonance,' 14th paragraph), how is the mind to catch the 
 rhythm or whatever it may be called, when the periods do not begin 
 and end at the same time, or, as we now say, when their phases are 
 not properly coincident? We shall not spend time discussing any 
 forms of the theory. There is nothing to discuss but mere speculation 
 or ignorance trying to "materialise" itself to knowledge. Ignorance 
 does not breed knowledge ; it is the waste land of science to be gradually 
 conquered by the shoots of knowledge that spread into it. Every 
 attempt to bridge the gap between vibration and sound must rest 
 upon greater success in the description of the physical process or of the 
 sounds themselves. For a complete study of either must finally lead 
 to the other, just as one real process binds the two into a single event. 
 In this case our way of success begins from the auditory side. 
 
 The line of progress is, in fact, continuous with the theory of tones 
 already developed. It is easily seen that if the upper tone fits so perfectly 
 into the lower tone in the case of the octave, it will do so, however 
 large the volume of the lower tone may be, so long as its volume is the 
 perfect fit. The lower tone may be moved gradually from the lowest 
 reach of the musical range of pitch till the upper tone reaches the 
 opposite extreme. What is common in this series will constitute the 
 interval of the octave as against its fusion or consonance. What is this 
 common feature? 
 
 At first glance there seems to be nothing that one can claim as the 
 basis of interval, since the balance or unity of the whole has been 
 allocated to the heard fusion. Even if this allocation was in the first 
 instance the outcome of a process of logical exclusion (77, 60ff.) it is
 
 40 THE NATUEE OF INTERVAL [CH. 
 
 confirmed by the kinship of the two terms thus brought into connexion 
 namely heard fusional unity or balance and conceptually formulated 
 balance or unity. A further process of discovery by exclusion seems 
 difficult in the case of the octave, because the fusional aspect of the 
 bi-tonal mass is here so prominent, both for sense and for conception. 
 Let us therefore consider a case from the lower grades of fusion. 
 
 There is only a slight difference between the major and the minor 
 thirds or between the different seconds and sevenths in the matter of 
 consonance or dissonance. If these bi-tonal masses had no other feature 
 than their fusion they would never have become so distinctive as they 
 now are in music. Then there must be some other feature in them that 
 provides a basis for our sense of interval. 
 
 Let us abstract for a moment from balance and unity altogether, as 
 if we did not apprehend it. Then we may make the following assertion. 
 So long as we were capable of noting the pitches and the volumes of 
 tones, even supposing they did not overlap (provided only they consisted 
 of a number of particles or 'spots' of sound, each ordinally distinct 
 and fixed, and so capable of being repeated precisely any number of 
 times), we should still be able to note the relative proportions of their 
 volumes and to construct to any given volume X a volume Y so that 
 their proportion should be the same as that of a standard pair P and Q. 
 We might not be able to do this as well as we judge and reproduce 
 intervals under present circumstances. Our margin of error would 
 probably be greater, just as it is when we compare the lengths or pro- 
 portions of visual lines from an unusual standpoint. We usually place 
 them directly in front of us and squarely to the line of sight. If we 
 could judge the proportions of volumes under these circumstances, it 
 must be evident that the comparison of the relative volumes of a pair 
 of tones is not made more difficult by the fact that the lower volume 
 consists partly of or includes the volume of the higher tone, so long 
 as the higher volume is distinguishable in the lower. In fact it may 
 well be easier ; for the volumes appear in the same place in the auditory 
 field. And the ease of comparison is increased by the facility with which 
 the volumes can be observed in succession. 
 
 In thus appealing to a sense of proportion we are merely giving 
 greater scope to a faculty of mind that experimental study has in recent 
 years shown to be of the greatest importance and of the finest efficiency. 
 If a visual standard of proportion is given, say two lines forming the 
 sides of a parallelogram, a fourth line can be constructed to a given 
 third that will show the same proportion with only a very slight error.
 
 vi] THE NATURE OF INTERVAL 41 
 
 The same sort of proportion can be carried through even with intervals 
 of time. 
 
 In any case since tonal intervals can, as a matter of fact, be so finely 
 learnt and reproduced as every musician knows, and since there is so 
 strong evidence that tones are volumes of sound of definite magnitudes, 
 consisting of a definite part of a fixed series of auditory particles, each 
 differing from the other only in its place in the series, we have every 
 right to claim that the real basis of our sense of interval is our observation 
 of a constant proportion between the volumes of tones. 
 
 This claim, though it has been won by careful theory, that carries 
 our minds through and beyond what the bare tones themselves suggest 
 to our simple observation, is in the end confirmed by our observation. 
 We have only to ask ourselves : is not what we call interval a constant 
 proportion between tones as we hear them? We shall perhaps not assent 
 at once if we merely observe a single interval reflectively. But take 
 that interval and think it successively on to a long series of tones of 
 different pitch. Such a test will show that we are in every way as fully 
 justified in translating sense of interval into sense of proportion as we 
 are in speaking of a sense of proportion in any department of experience 
 at all. We are in the sense of interval finding the proportions of things 
 that really bear proportion to one another, and we do so very accurately. 
 
 When we establish relations of proportion between lengths of line 
 by mere visual inspection of them, what do we do? And what are we 
 aware of? We inspect these lines and compare them as to their lengths 
 which appear as sensible magnitudes. We base our judgment of propor- 
 tion upon this inspection. We are aware of the magnitudes we inspect 
 as lengths and we feel keenly whether a known or given standard of 
 proportion is repeated in a given pair of lines, making thereby in our 
 judgments only a very small margin of error. In judging the proportions 
 of tones, or in judging tonal interval we do exactly the same. We inspect 
 tonal lines of a little breadth, or, as we usually call them, tonal volumes. 
 In these volumes pitches appear, not detracting from our power to 
 judge of interval, but rather aiding it considerably by giving it a sort 
 of focus. We are aware that the tones we compare have volumes or 
 that the whole volume constituting an interval has a particular volumic 
 figure. We are aware of this even though we could not describe what 
 we are aware of in the clear conceptual terms we readily use in vision. 
 For in vision we are all both in practice and in theory highly expert, 
 whereas in hearing most of us are in practice very inexpert and we have 
 all been devoid of proper theoretical insight. Now that the insight has
 
 42 THE NATURE OF INTERVAL [OH. 
 
 come, we can see that it gives a true description of what we do and of 
 what occupies our attention while we estimate interval. In judging 
 interval we also feel keenly whether a known or given standard of 
 proportion is repeated in a given interval and the margin of error made 
 by expert judges is very small. 
 
 The study of these lower grades of consonance as intervals shows 
 us, moreover, that we can fix any interval as an interval in our memory. 
 The interval may be 24 : 31 equally as well as 24 : 32, provided it be 
 fixed in the memory by frequent repetition and attention. Of course 
 it is much easier to learn the consonant intervals because they have a 
 special attraction for the attention and for the memory. For on the 
 one hand they fuse, and on the other they are few and important. 
 Intervals, such as the tritone and the major seventh, which differ only 
 a little in size from some prominent consonance, are hard to sing because 
 they tend to slide into the easy consonance, as it were. But with sufficient 
 practice any such difficulty may be overcome. As Alfred Day said in a 
 similar connexion : "Practice is for the purpose of overcoming difficulties 
 and not of evading them" (10, 7). 
 
 It is conceivable, as some have claimed, that those who constantly 
 practise with the intervals of equal temperament should finally come 
 to use them and to think in them by preference 1 . When the circumstances 
 of judging are most favourable, the accuracy with which deviations 
 from a familiar interval can be detected is very great. Thus Meyer 
 and Stumpf got collective results showing inter alia an accuracy of 
 74 % for a deviation of 0-78 vb. from an ascending major third 
 (600 vbs.); +2-18 gave 72%. An individual result (Stumpf 's) gave 
 88 % for - 0-78 from the ascending third and 82 % for + 2-18 (73, 
 
 358 ff.). 
 
 This process of abstraction has thus yielded us a new feature of 
 complex volumes, namely the proportion of their parts or interval. 
 We may now look back upon the well-balanced fusions that seemed to 
 offer no other feature for analysis than their obvious balance, and 
 reconsider the problem. 
 
 The octave, we may say now, is not only a special fusion; it is an 
 interval as well. The tones that form it do not only fit peculiarly into 
 one another, but they also bear a certain volumic proportion to one 
 another. Thus we have a double basis by which to fix the octave in 
 
 1 This sentence bears no reference to the controversy on the respective merits of 
 equal and just temperament.
 
 vi] THE NATURE OF INTERVAL 43 
 
 the attention and memory and a double use for it in music. If we ask 
 what are the respective contributions of fusion and of interval to the 
 importance of the role played by the octave in music, there can be 
 no hesitation as to the answer. By far the more important aspect of 
 it is its fusion. Had we not a linear field of sound, but an areal one, 
 as in vision, in which tones could be given at varying distances from 
 one another without overlapping at all, we should have attached as 
 little importance in music to a 1 : 2 proportion between tonal volumes 
 as we attach to that proportion between the lengths of lines in visual 
 art. The 1 : 2 proportion stands forth in music because the ' upper ' 
 ends of all tones are identical and tones overlap completely from thence 
 'downwards.' It is idle to speculate as to what kind of music we might 
 have made if we had had such an areal, or even a cubic, field of sound. 
 I mean, of course, areal for musical purposes. It is areal already, as 
 shown above, as a whole, but the transverse dimension has no musical 
 utility. 
 
 Interval might well be called our sense of form in sound, when 
 fusion would be our sense of mass, as it were. There is no use in labouring 
 these analogies between sight and sound, except in so far as they help 
 to bring out the underlying identity of structure in the two senses, 
 and so to understand the nature of each better. Still less should we 
 attempt to base practical reforms or advances upon these interpretations 
 by trying to raise upon the foundations of tonal mass or tonal form 
 structures analogous to those developed in the visual arts upon the 
 foundations of those names. If such structures are naturally possible 
 to music, they will probably have been created to some extent already. 
 If the analogies suggested are real, the first event to follow may well 
 be the discovery that certain types of music differ by their emphasis 
 upon fusion or upon interval, upon mass or upon form. 
 
 Possibly that is the real meaning of the great difference of nature 
 and view between harmonic and polyphonic music, the former being 
 the art of mass of fusional (consonantal) effects, of course. No visual 
 art is purely a construction of masses or of forms alone. It is impossible 
 to separate mass and form in this way. Every mass must have a form, 
 and every form that is at least bi-dimensional indicates mass to some 
 degree or other. Visual arts do, nevertheless, differ in the relative extent 
 to which they build on mass and form. Similarly in music. Every 
 fusion has a form and every interval has some degree of balance or mass 
 unity about it. But polyphonic is commonly said to differ from harmonic
 
 44 THE NATURE OF INTERVAL [CH. vi 
 
 music in that the one is viewed horizontally, the other perpendicularly; 
 or in that the one regards chords rather as a whole, while the other 
 takes more interest in creating and following out the lines, as it were, 
 that run side by side throughout the successive groups of sounds. The 
 difference is one of degree. We shall see as we proceed, that this dis- 
 tinction of attitudes towards groups of tones is of the greatest importance 
 for a study of the foundations of music.
 
 CHAPTER VII 
 
 THE MUSICAL RANGE OF PITCH 
 
 ONE of the most curious facts of hearing is that music is restricted to 
 a certain range of pitch. Outside the limits thereof it is no longer 
 possible to make music. The pianoforte makes these limits familiar 
 to every one. The lowest tone on the large concert grand piano is A 2 , 
 the highest is c 5 . These pitches include a little more than seven octaves. 
 Anyone may notice upon the piano how the lowest notes seem to give 
 an insufficient difference of pitch from their neighbours. The intervals 
 of a major second seem too small; those of a minor second seem to be 
 hardly distinguishable as intervals. A little more, one thinks, and the 
 two tones would seem to be the same. At the upper end of the keyboard 
 a similar change is noticed, though it is not nearly so distinct upon the 
 piano. But it appears clearly if we carry the pitch of tone physically 
 some distance into the c 5 octave. 
 
 This limitation of range does not depend upon any purely physical 
 restriction. Periodic waves can be produced below and above these 
 limits and pairs of tones maintain their proper physical relations to 
 one another unchanged. Nor does the phenomenon seem to depend 
 upon an incapacity of the ear to hear tone. For the ear responds with 
 a tone-like sound to physical rates of vibration at least four or five 
 times as great as that required for c 5 . A great deal of patient effort 
 and ingenuity has been spent upon the attempt to fix the vibrational 
 limits of hearing as accurately as possible. They will always be uncertain ; 
 for their physical sources are not only hard to control and to gauge 
 correctly, but the range of hearing varies considerably from person 
 to person and from youth to age. It also varies considerably with the 
 intensity of the physical stimulus. In fact it is possible that within 
 a large range of physical differences any rate of vibration will produce 
 some auditory effect or other, and if loud enough it may be a tonal 
 effect, without there being any real differences in these effects, except 
 minor or accidental ones. The determination of the limits of hearing 
 would thus be illusory, after a certain point. We shall realise this 
 possibility better further on. 
 
 Material has been gathered carefully by experimental means towards
 
 46 THE MUSICAL RANGE OF PITCH [CH. 
 
 an adequate description of the limits of the musical range of tone. It is 
 found that no sharp boundaries mark it out. Towards the upper side 
 it shows itself first in a slight apparent flattening of the pitch of tones 
 from what the rate of vibration of its source leads us to expect. 
 Gradually as the tone is raised this flattening increases to a semitone, 
 and even to a tone. Beyond this point the estimation of pitch soon 
 breaks down altogether. A similar gradual deterioration of judgment 
 is found on the lower side, but here the very low tones seem to be 
 sharper than they should be, according to the known physical rates 
 of vibration. 
 
 If interval is constituted by constant proportion of volume, it follows 
 that so long as the pitch of a tone of these high or low regions can be 
 estimated with confidence and regularity, there is at least on the 
 phenomenal side or in the tones themselves nothing amiss. The octave 
 is still in every way an octave for hearing. The discrepancy lies only 
 between the auditory and the physical series. It requires a greater 
 ratio of physical vibration to produce a volumic octave in the high 
 border region than is usual in the middle range of hearing. The ratio 
 is 1:2+, instead of 1:2. Similarly in the low border region the 
 physical ratio downwards is 1 : | instead of exactly 1 : |. On the 
 higher side there is therefore evidently some difficulty in making the 
 volume of tone smaller in the usual proportion. So the rate of vibration 
 has to be increased a little in order to get a reduction of the volume 
 by half exactly. On the lower side there is evidently a difficulty in 
 making a volume of the usual large size. The sizes required are so great 
 that a reduction of the rate of vibration beyond the half is necessary 
 in order to get precisely the double volume. 
 
 These special difficulties receive an easy explanation by reference 
 to the physical sense-organ. The cochlea is quite a small thing, and 
 although its functions seem to be remarkably independent of its size, 
 they can be so only relatively, not absolutely. There must come a point 
 at which the organ will fail to respond properly to a very short wave- 
 length of vibration. Similarly it will be at some point or other finally 
 unable to accommodate the great long waves of sound. No apparatus 
 at all will cover an infinite range of forces. It will fail to work beyond 
 certain extremes, and towards these it will lag behind the change of 
 force applied. At first this will be only a perceptible lag, finally no 
 further change will be given. The ear will respond with the extreme 
 possible to it on either side. 
 
 This seems perhaps to be the case at the upper pitch limit of hearing.
 
 vii] THE MUSICAL RANGE OF PITCH 47 
 
 For a considerable period beyond the end of the musical range tones 
 are still heard. They become slowly thinner and sharper and finally 
 disappear gradually into a mere hiss or puff. At the lower end the longer 
 waves seem after a time to produce no more effect upon the ear at all. 
 At the most they are felt as puffs of air against the drum of the ear. 
 
 The musical range of hearing, then, is the range within which the 
 change of tonal volume keeps march with the change of vibratory 
 rate. As long as this holds good, instruments may be constructed and 
 played with freedom and with complete certainty as to the musical 
 effect upon the listener. It is, of course, conceivable that musical 
 work could be carried up to the outer limits of the border region for 
 a single listener at least. But the physical ratios required at these 
 last points in order to maintain the desired relationships of tonal volume 
 would not be generally valid. They vary considerably from person to 
 person. Consequently music is more or less obliged to discard these 
 border regions in so far as precise effects are desired. Thus the musical 
 range of tone becomes the range within which the changes of volume 
 and of vibratory rate are exactly inversely proportional to one another.
 
 CHAPTER VIII 
 
 OUR POINT OP VIEW TOWARDS THE AUDITORY FIELD 
 
 WHEN the physics and physiology of vision had advanced far enough 
 to understand roughly the build and functions of the eye, it appeared 
 evident that the impression cast through the lens of the eye upon the 
 sensitive surface was, like that seen on the ground- glass focusing- 
 plate of a camera inverted. To many men this seemed an extraordinary 
 fact. It yielded for their minds a fundamental problem : to show by 
 what means the image of the eye was turned back to its proper orienta- 
 tion. For we do not see things upside down at all. Consequently either 
 the retinal image must be so transmitted to the brain as to arrive 
 there right side up, or the soul itself must give us a properly adjusted 
 view for the inverted impression it receives. Echoes of this kind of 
 reasoning may be found in books of no distant date. 
 
 One philosophic answer to this problem seemed to make it ridiculous. 
 That was the claim that it was here a question not of absolute, but 
 only of relative positions. As all our vision is 'inverted,' none of it is. 
 If the whole world expanded suddenly threefold or if time shrivelled 
 to twice its present rate, we should none of us be aware of the change. 
 So it is a matter of indifference whether visual impressions reach the 
 brain erect or sloped or inverted, so long as they are all modified in the 
 same way. For all we know they may be distorted in the strangest 
 ways in the process of being accommodated to the zig-zag turns of the 
 cerebral convolutions. A certain eminent writer has even pointed out 
 that in our field of vision there is no trace of the holes and slits in the 
 neural continuity that must correspond to blood vessels and connective 
 tissue and such like. He considers this to be an anomalous feature in 
 any systematic co-ordination of brain and mind. 
 
 The philosophic ridicule of this problem of inversion is just and 
 proper as far as it goes. In the first instance, or primitively, as it were, 
 it is a matter of indifference how the visual field is orientated, if indeed 
 it can be said to have an orientation to anything outside itself at all. 
 The difficulty of any such absolute orientation may be illustrated in the 
 terms of popular metaphysics. That supposes very often that besides 
 body and mind we have a soul. And perhaps the soul has, or possesses,
 
 CH. vin] THE AUDITORY FIELD 49 
 
 the mind. In the opinion of some it is the soul that gives us what we 
 find 'in our minds,' that is to say, our experiences. The body somehow 
 acts upon the soul and the soul responds in its own unique and scien- 
 tifically incomprehensible way. If so, then what is the exact place 
 occupied by the soul? Is it in the brain, or at the brain? The question 
 seems absurd to some. They say the soul has no place. It can even 
 be acted upon from two places at once, e.g. from the two eyes or the 
 two ears, and it then responds by giving us unitary experience. But, 
 nevertheless, for all we know the soul might be far away from the body, 
 say in the star Sirius, so long as an arrangement had been made 
 whereby it should be acted upon by the particular human body on our 
 planet that belongs to it. The problem of the absolute orientation of 
 the visual field is just as insoluble as this problem of the soul's distance 
 from the body. Vision has no absolute orientation to anything that 
 could ever be discovered by us. 
 
 But has it not an orientation towards the soul? Suppose you invert 
 the printed page your eyes are now fixed upon, and try to read it. If 
 your soul may not be disconcerted by the change, your faculty of reading 
 will surely feel a difficulty. The disturbance is almost as great when 
 the page is turned through a quarter circle. One of the devices often 
 used in order that differences of colour may be more striking and clear, 
 is to bend down so that the head is inverted and to view the object 
 or the landscape from this unusual position. We may even arrange in 
 this way that nothing in the whole field of vision remains uninverted, 
 or visible in its uninverted relation, not even our own cheeks and eye- 
 brows. Whatever may have been the case in the absolute beginning, 
 there can be no doubt that we, brain or soul or both, do get accustomed 
 to one mode of presentation. 
 
 An American psychologist threw much light on this question by 
 wearing for many days in succession glasses that inverted the whole 
 of his visual field. "The first effect was to make things, as seen, appear 
 to be in a totally different place from that in which they were felt. 
 But this discord between visual and tactual positions tended gradually 
 to disappear; not that the visual scene finally turned to the position 
 it had before the inversion, but rather the tactual feeling of things 
 tended to swing into line with the altered sight of them. The observer 
 came more and more to refer his touch impressions to the place where 
 he saw the object to be; so that it was clearly a mere matter of time 
 when a complete agreement of touch and sight would be secured under 
 these unusual conditions. And when once the sight of things and the 
 
 W. F. M . 4
 
 50 OUR POINT OF VIEW [CH. 
 
 feeling of them accord perfectly, then all that we mean by upright 
 vision has been attained" (63, 147). 
 
 From this important experiment, so trying to the patience of the 
 experimenter, we must infer that, even if the visual field has no absolute 
 orientation, it has at least a correlation with the other sensory fields. 
 Visual 'up' is connected in our minds with muscular 'up,' and so on. 
 
 But there must be still more than this. The particles or minimal 
 spots of which any sensory field may properly be held to consist, are 
 both absolutely and relatively different from one another. This difference 
 applies to their ordinal attribute. The particles of sight we call 'up,' 
 are ordinally what they are; they have an absolute differentia inherent 
 in them. This order of theirs we connect by association with a muscular 
 particle (or a series leading thereto) which has also an absolute 'order' 
 of its own. But should circumstances suggest it, we are free to change 
 this connexion by association, so that another visual particle, ordinally 
 very different, will come to be correlated with the muscular 'up.' And 
 so on. Not that these orders are, as it were, absolute places in the 
 universe. But they cannot be called merely relative, because it is not 
 thinking alone that gives them their order towards one another. They 
 come to our thought already ordered; they are already such that if 
 and when we gather them together, we shall see that they actually form 
 a system. That is, their order is inherent in each of them ; it is absolute. 
 
 But this does not prevent us from thinking these orders in relation 
 to one another and abstracting from their absolute basis or from the 
 associations that rest upon the latter. We can turn a triangle or a square 
 about in the visual field so that it takes up almost any sort of orientation 
 within or upon the absolute constituents of that field. And so we learn 
 to think a square and a triangle independently of its orientation. But 
 if we do not have occasion to make these variations, we shall learn 
 to see a figure and even to recognise it best, or perhaps only, when it 
 is placed a certain way up. Sometimes we cannot easily make these 
 variations. In other cases there are advantages in avoiding them; for 
 one and the same figure from the point of view of that figure only 
 may become several figures, if presented in certain fixed orientations 
 and associated with different meanings in each case. Thus h and y 
 (as written) are almost inversions of one another, and yet they are used 
 as signs of different sounds. So are many other pairs of letters. The 
 advantages are here all in favour of letting the absolutist tendencies 
 of visual orientation prevail.
 
 viii] TOWARDS THE AUDITORY FIELD 51 
 
 Now all this kind of thing may be in general quite familiar in vision. 
 In hearing, however, where something similar occurs, both the facts 
 and their explanation are probably much less familiar. 
 
 The pitch of a chord that is perfectly stationary and is not at the 
 moment apprehended as part of a melodic sequence, is most frequently 
 felt to be the pitch of its lowest component, even when that is not the 
 strongest. The attention seems to fall most easily upon the lowest 
 tone. It may certainly be directed by melodic means or by voluntary 
 effort upon any other component of the whole sound, whether that be 
 a primary tone or an upper partial or a difference tone, or the like. 
 But left to itself and unguided it falls back upon the lowest component, 
 if it is not too weak. It will even fall without instruction upon the 
 lowest difference-tone that may be present. 
 
 Amongst the ancient Greeks it appears that the instrumental 
 accompaniment was always above or higher in pitch than the melodic 
 voice. "In the twelfth problem it is explicitly stated that the voice 
 occupies the lower part of the harmony. 'Why does the lower of two 
 notes always take up the melody? 1 ...' There are extant in Plutarch 
 two texts no less decisive of which the first is : 'What is the cause of 
 consonance and why, when consonant sounds are struck simultaneously, 
 does the melody belong to the lower?' And the second : 'In the same 
 way as, if two consonant sounds are taken, it is the lower that makes 
 the song.' The custom of putting the accompaniment higher seems 
 to have maintained itself during the Roman period" (14, 364).. The 
 accompaniment might descend to unison with the voice, but not go 
 below it. In early Western music the vox principalis was at first higher 
 than the vox organalis, but after a time it took the lower place and 
 remained there (81, 96). 
 
 This constant obviousness of the lowest tone is an important factor 
 in music, where, as Macfarren said, the bass "is always the most 
 sonorous part in the harmony" (35, 99). It is to be explained, as 
 already indicated, by the fact that the lowest tone includes all simul- 
 taneous higher tones within its volume, and that the pitch of the 
 lowest component is the central point of the whole tonal mass of any 
 moment. Tone is specifically balanced or graded volume of sound. 
 In so far as we apprehend sounds tonally at all, we must look at 
 
 1 Instead of "melody" Sturapf understands in the first place "pitch" (65, 19). Compare 
 the suggestions raised by my conclusions below, Chap, xvni, end. In connected music 
 the more prominent pitch of an isolated interval would become the more prominent 
 melody. 
 
 42
 
 52 OUE POINT OF VIEW [CH. 
 
 them centrally, as it were. Thus whatever else we may observe in a 
 tonal mass, if we apprehend it as a whole, we shall inevitably look at 
 it centrally and so most readily come upon the pitch of its largest 
 volume, i.e. of the lowest component. 
 
 This prevailing attitude shows itself in a number of other ways. 
 We agree in reckoning the interval between any two pitches upwards, 
 unless some special indication to the contrary is given. Thus C-E is to 
 be taken as a major third, not as a minor sixth. Even the Greeks, 
 who in practice found it more fitting to pass from high to low than 
 conversely, reckoned all intervals theoretically from below upwards 
 (16, 89; 14, 173 ff.). Rising of pitch gives us the impression of departure, 
 lowering of pitch that of approach. We incline to take a scale from 
 below upwards and back again, rather than downwards and up again 
 to the starting-point. In a major chord we consider the major third 
 as the first of the two intervals, the minor third as the second. The 
 tonic of a major chord, whether its component tones are given succes- 
 sively or simultaneously, is held to be the lowest of the three. The 
 attempt has been made to look upon the highest tone of the minor 
 triad as its root or tonic. But the very strangeness of the claim, apart 
 from the validity of the special arguments advanced in its favour, 
 shows that it does not correspond to our actual attitude towards the 
 chord. Moreover, when an interval is mistaken for a single tone, as in 
 the experiments on fusion, the pitch ascribed to it is in the majority 
 of cases that of the lower tone. 
 
 Another strong evidence of this central attitude to tonal groups is 
 found in certain striking differences between ascending and descending 
 intervals involving the same tones. "The most of those who can 
 recognise intervals at all have learnt in the first place to judge them 
 in the ascending form. The estimation of descending intervals is much 
 harder; in fact it is primarily quite a different task.... The difficulty 
 of judging descending intervals appears not only amongst less practised 
 observers, but also amongst... persons who had all enjoyed a good 
 musical education. In judging descending intervals indirect criteria 
 were often used by them. The time spent in recognising these intervals 
 was also larger than that required for ascending intervals" (37, 192). 
 
 This may seem at first to be a very extraordinary fact, hardly 
 creditable by those who recognise all intervals at once without hesitation 
 or by those who find difficulty in naming any by ear alone. It is certainly 
 incompatible with any purely relativistic interpretation. For if the 
 relation a to 6 is recognisable, the relation of b to a should be so also
 
 vin] TOWARDS THE AUDITORY FIELD 53 
 
 as a matter of course, since it is the same relation. But if a point of 
 view has been adopted and if a and 6 are not of a purely qualitative 
 nature, we can readily understand that the appearance of a-b from 
 the standpoint of 6 may be very different from its appearance from 
 the position a. The face of a friend seems very strange when it is seen 
 inverted. Even the letters of our alphabet or simple ornamental figures 
 become unfamiliar then. 
 
 This peculiar difference produced by the direction of interval is 
 seconded by a similar distinction between simultaneous and successive 
 intervals. When musically untrained persons have been taught to 
 attach the correct names to simultaneous intervals, it is found that 
 they are quite incapable of naming the corresponding successive forms, 
 in spite of great practice at the former task. They cannot mentally 
 convert the succession into simultaneity (37, 192). 
 
 We may therefore look upon it as well founded that we do adopt 
 a strangely prevailing attitude towards the tonal series. Our point 
 of view is for any moment that of the centre of the whole tonal mass. 
 This centre need not, of course, merely be the centre of a single tone 
 or of the momentary mass of sound that is at the ear. It may be the 
 centre of a tonal complex begun a moment ago and lasting on for a 
 span of time till it is completed. How long this span may be, will 
 depend greatly upon our musical practice and upon the musical coherence 
 and stability of the complex that is presented. These complexes vary 
 in length and complexity very much. It is also possible that in viewing 
 the various parts of this complex as they flow past us, we do not need 
 to maintain the central position that is valid for the whole complex 
 in any rigid way, so long as our disposition towards it remains ready 
 and active. We may then wander about with the centres of each 
 momentary sound mass, always having it in our power to see the 
 relation of that to the general centre of the whole and to return to the 
 latter if required. All this would, of course, not be an inevitable and 
 unshakable consequence of the primary centrality of tone, but would 
 gradually develop out of it by the practice and mental skill of the 
 listener and by the support given to him by the devices of musical art. 
 Thus we see how the primary point of view towards tone might develop 
 towards the special point of view we know in music as tonality, the 
 feeling for a tonic, a point of reference for the tones and chords of a 
 musical unit. 
 
 Nor does the fact that the central point of view towards tone is 
 the natural and prevalent attitude prevent us from acquiring another
 
 54 THE AUDITORY FIELD [OH. vm 
 
 point of view by special practice or preference. Those who sing a 
 certain part in songs or hymns regularly and whose musical practice 
 is predominantly of this kind, will doubtless find it easier to follow 
 their usual habit. The musical analysis of many persons is confined 
 to attention to soprano melodies. Even if their analysis goes beyond 
 this, their greatest practice and interest may yet tend oftenest to the 
 highest voice, so that if a single chord be given they will select 
 from it as its pitch its highest (primary) component. Modern music 
 teaches everyone to pay special attention to soprano melody. For it 
 commonly endeavours to put the maximum of interest into one such 
 melody and subordinates the melodic interest of other voices to their 
 harmonic beauty. D. F. Tovey expresses this when he defines melody 
 as "the surface of music" (75). Our present interest is not to investigate 
 or to depreciate the importance of any such special points of view 
 towards music, but only to show how various facts indicate that the 
 natural, original or fundamental point of view is a central one, or 
 extends from a variable point or centre upwards in the tonal range. 
 
 At the same time these facts support strongly the theory of the 
 volumic proportional nature of interval and bring the study of tone 
 and music into most intimate agreement with facts that better natural 
 endowment, the greater scope of physical variation and greater practice 
 have made more or less familiar to us all in vision.
 
 CHAPTER IX 
 
 THE RELATIVE IMPORTANCE OF SYNTHESIS AND OF ANALYSIS 
 
 WE have now gone so far as to be able to look back upon the field 
 of tone and to survey it somewhat as a whole. 
 
 The opinion has been often expressed that science can never give 
 a proper account of any art, because the aim of science is analytic; 
 it strives to dissect and to divide, tracing each part to its separate 
 root and origin. It must necessarily lose the life and spirit of the whole. 
 No doubt this is true so long as a science is busy over the preliminary 
 efforts of analysis and has not yet reached the stage of tracing the 
 synthesis that binds the many parts together. But analysis is not the 
 final condition in which scientific results are to be left. 
 
 The study of the body involves a long course of special study of 
 each distinguishable part and of its own particular functions. It is 
 only thus that we can learn what primary functions or processes are 
 at work in the living body. And it is only from the basis of this knowledge 
 that we can venture to explain the united work or the integrative 
 action of the living body. 
 
 The study of the mind at first calls for a minute examination of 
 every distinguishable experience, its fundamental variability and its 
 primary relations to other experiences. But the science of the mind 
 is not to be taken as a mere catalogue of pieces and processes without 
 connexion with one another. That would be to mistake its achievements 
 at a certain early period of its development for the results it may in 
 the course of time properly expect to attain. One of its duties is to 
 aspire to show how the mind of the average man appears to him as 
 it does and why. 
 
 In the same way the science of music has first to dig down to its 
 foundations and show their form and connexions. Only then can it 
 build upwards from these and aspire to give a full and true account 
 of music as it appears to the musical mind, i.e. to the mind that is not 
 crowded with scientific knowledge concerning music and actually 
 thinking of it, but to the mind even though it be the same mind or 
 person that is for the moment hearing and enjoying music in the 
 ordinary way.
 
 56 THE RELATIVE IMPORTANCE OF SYNTHESIS [CH. 
 
 The science of music has for centuries paid the greatest attention 
 to harmonics or upper partial tones. It has tried to explain many 
 things by them. The insufficiency of the results has turned the hopes 
 of theorists in later years towards the lower tones that appear in chords, 
 namely to the difference-tones. But of all these things the ordinary 
 musical mind is quite regardless in hearing and enjoying music. It is 
 only with difficulty and effort that it can be brought to recognise their 
 existence even when the attention is not aesthetically engaged. And 
 when it is again so occupied, harmonics and difference-tones disappear 
 from view entirely. That fact alone suggests the view that harmonics 
 and difference-tones have not the central importance for musical theory 
 that has often been claimed for them. And it confirms a theory that 
 can find other foundations of greater validity. 
 
 This does not, however, mean, as some have seemed to think, that 
 the scientific attention creates harmonics, or that harmonics come and 
 go according to the inclination of observation. I say 'seemed to think,' 
 for no one can venture to maintain such a view outright. We may 
 want for our satisfaction to think that "the self sets itself" first and 
 then all the rest of the world, including harmonics, according to its 
 inclinations of self-realisation. If it is possible to leave this marvellous 
 power to a Universal Self, we may well do so. But for our own self we 
 must refuse to believe that it is able by mere change of attention to 
 set anything into being at all. If harmonics are there when we attend 
 to them, then they are also there when we do not attend to them. 
 What we have to explain is why, when we attend to them, they appear 
 in a different way than they otherwise do. 
 
 And that is not a difficult task. When we attend to a harmonic, 
 we concentrate our inward gaze upon it alone to the exclusion of any 
 setting or circumstances it may stand in. So we notice its own particular 
 pitch and we can form a fairly sufficient estimate of its own particular 
 volume. We may arrange for the independent production of a tone 
 very like it and, by noticing the beating of the latter with the harmonic, 
 estimate its pitch precisely. But we do not create it by our attention. 
 For we have no knowledge from our will alone what its properties will 
 be and real tones will not beat with fancied ones. 
 
 When we cease to attend specially to harmonics they are in them- 
 selves quite unaltered thereby. But if we then attend to the tone that 
 contains them, we hear them in their full setting; we hear them as 
 a part of the tone or chord we are attending to. And then, as everyone 
 knows, they appear to us as the particular blend (or timbre) of the tone.
 
 ix] AND OF ANALYSIS 57 
 
 They do so because, being higher than the primary tone which gives 
 the whole tone its musical pitch, they all fall within its volume. And 
 in good musical tones, the upper partials are of a restricted intensity, 
 wherefore they do not stand out prominently in the volume of the 
 whole tone so as to call the attention specially to themselves. They 
 leave the balance and symmetry of the fundamental still obvious. 
 These qualities are no longer so perfect, of course, as are those of the 
 pure tone. But they are far from being so vague and deteriorated that 
 the sound could be mistaken for noise, in which balance and symmetry 
 have been lost or at least made very hard to find. 
 
 The harmonics of a good musical tone only make a slight change 
 of surface, as it were, in the whole tone. It no longer remains perfectly 
 smooth like the pure tone; its volume acquires a character whose 
 nature depends upon the harmonics present. A set of very high harmonics 
 will give the tone a touch of highness or brightness. The lower harmonics 
 will give more variety to the central body of the tone; it will not be 
 empty and poor, like the pure tone, but full of interest and rich. If 
 only the uneven numbered partials occur, the tone will take on another 
 character, one that appears in the sounds produced by the nasal voice 
 and by hollow cavities of various kinds, so that we associate the idea 
 of hollo wness with it, and call it a hollow sound. And so on. 
 
 The interests of music are not commonly served by sounds in which 
 partials attract the attention to themselves or are separately distin- 
 guishable with ease. Those tones are the most valuable in which the 
 minimal reduction of smoothness is compensated by a maximal richness 
 and interest of pitch-blend. Tone must be rich and strong without 
 being rough, and smooth without being dull or poor. It should be at 
 once as full and as rich as may be. In analytical terms, the fundamental 
 must be present in good strength to give the tone a fullness of the volume 
 it 'aspires' to or is meant to be; and a typical series of partials should 
 'colour' it or give it a characteristic surface without being so strong 
 as either singly or collectively to outweigh the fundamental or to stand 
 forth in it so much that they take upon themselves the rank of primary 
 sounds tones actually played separately by the performer and written 
 by the composer or intended to be heard separately. 
 
 It is unnecessary to recall that great variety of beautiful tone- 
 surface is of the highest importance in music. These blends give a new 
 interest to repetition and a new line of variation by which the hearer's 
 mind may be led to give ear to the secret of the soul's life that the artist 
 strives to convey.
 
 58 THE RELATIVE IMPORTANCE OF SYNTHESIS [CH. 
 
 The musical attitude towards harmonics, then, is the synthetic 
 attitude. They create beauty when their synthesis is easy or inevitable, 
 i.e. when their strength is so graded and unobtrusive that they appear 
 to the attention only as a minor modification of the tonal volumes 
 that compel the attention. But when the attention is used analytically 
 as when we pass from stone to stone of an architectural surface or from 
 stroke to stroke of the brush in a picture harmonics can be inspected 
 singly. The rest of the tonal mass tends to disintegrate. The attention 
 is then concentrated on the part and is scattered in the rest; whereas 
 in a synthetic unity such as an artistic object the centre of attention 
 is so placed that it radiates easily to the parts and binds them together 
 in itself, while they point towards it and so make it easy to find rapidly. 
 
 Let us now consider difference-tones in the same relations. These 
 lower partials, as it were, do not accompany single tones, so that they 
 cannot play the same part in giving blend or surface to a tone as upper 
 partials do. They appear only when at least two primary tones are 
 sounded. Of course, they must also be produced by the interaction of 
 upper partials as primaries. But the artistic subjection of these to the 
 fundamental partial does not allow of their usually appearing in a single 
 blended tone in any noticeable degree. Difference-tones are in any 
 case weaker than their primaries, so that if they originate from the 
 partials of a blend, they will be weaker than these and will therefore 
 have less chance of being separately noticed in a blended tone than 
 have its partials. Besides, the partials of the harmonic series could 
 never produce either a difference-tone that was lower than the funda- 
 mental of that series or that did not coincide with some member of 
 the theoretical harmonic series. So any new component of a sound 
 that was produced as the difference-tones of its partials would only 
 appear to be another partial of that sound. 
 
 The difference-tones that are due to primary tones are of quite 
 considerable strength. This is true at least for the first difference-tone 
 (higher rate of vibration minus the lower, or h I) and for the second 
 difference-tone (21 h). The other difference-tones are much weaker 
 and very difficult to hear, so that their inaudibility to the ordinary 
 ear under usual circumstances hardly forms a problem for any possible 
 theory of sound. In spite of their loudness these first two difference- 
 tones are much less easily noticed than partials as the much later 
 discovery of them shows. Reasons for this obscurity are not far to seek. 
 The strongest is the nature of their origin. They appear only when
 
 ix] AND OF ANALYSIS 59 
 
 two sounds are played together. In order to detect them one has to 
 consider carefully what exactly each of these two sounds separately 
 contains and then to subtract the sum from what is heard when both 
 are played together. Most persons do not trouble to do this. They 
 expect the tonal mass of the two sounds to contain more than that 
 of either, as it obviously does. But the peculiar overlapping and blending 
 of simultaneous tones prevents them in most cases from discerning 
 precisely the exact contribution of each of the primary components. 
 Thus the new whole passes as the peculiar mixture of the primaries. 
 This tendency is encouraged by the fact that in the octave where the 
 fusion of the primaries gives the simplest product, there is only one 
 difference-tone which is identical with the lower primary. So then 
 where one might most readily have detected an addition, there is nothing 
 new to find. And in the fifth, the only difference-tone is exactly an 
 octave below the lower primary and so fuses with it as much as any 
 tone could, thus making detection again difficult. A third reason lies 
 in the fact that difference-tones are not under ordinary circumstances 
 found to exist outside the ear, so that they could not be discovered, 
 as were so many of the chief facts of acoustics, from a study of the 
 movements and resonance of the sonorous body. They had to be 
 found purely by the inspection of sound itself. And the attention was 
 naturally directed in that to the primary sounds that were intended 
 and played and upon which musical structure primarily rests. 
 
 Nevertheless difference-tones did not, of course, first come into being 
 at their discovery. They were there all along, moulding the character 
 of chords as a whole, giving them especially a touch of a largeness of 
 volume that their primaries did not contain. The listener, as is often said, 
 is unconsciously affected by them. That does not mean, to be sure, 
 that his body or brain is affected by them, but not his mind. It means 
 only that while the lowness is there in his sensations in a particular 
 form, and he hears it as a lowness appertaining to the whole sound, 
 yet he does not separate it out in its discrete form in the whole and 
 know it as such. It would, therefore, be better to say that the listener 
 is unwittingly affected by the difference-tones. For if sensation is a 
 form of consciousness, and the difference-tones are in sensation, the 
 hearer is of course conscious of them. If sensations are considered to 
 be objects presented to the mind, which is only conscious of them 
 when it knows them individually, then the hearer is unconscious of 
 difference-tones even when he attends to the sound complex in which 
 they appear before him, until he has separated them out from the group
 
 60 THE RELATIVE IMPORTANCE OF SYNTHESIS [CH. 
 
 of sensory objects presented to him and has cognised them individu- 
 ally. 
 
 Having thus surveyed the outlying components of a sound mass, 
 we may now deal with the relative importance of synthesis and analysis 
 amongst the primaries. 
 
 It is to be noted first, however, that the objection brought against 
 harmonics and difference-tones as the foundations and regulators of 
 musical structure does not hold for our interpretations of the primaries. 
 That objection is that the composer and the hearer commonly know 
 nothing about harmonics and difference-tones and care still less, least 
 of all when they are actually in the aesthetic attitude. They neither 
 know of these things nor do they attend to them. But, while they have 
 certainly not known about the volumes and coincidences and proportions 
 of tones either, they have indeed always attended to them. For tone 
 and interval are not derived from volume and proportion as from things 
 that lie in a land beyond their own : on the contrary, they are volume 
 and proportion. Whoever attends to tone and to interval, attends to 
 volume in its balance and symmetry and to proportion of volumes. 
 What our theory has done is neither to trace the heredity of tone and 
 of interval and of fusion, nor to say what remote stars have influenced 
 their horoscope; but it has dissected the very body of these things, 
 as it were, showing what they consist of and how they are related to 
 one another and to other similar things. Thus the change for the 
 artist and hearer is merely from the practical and aesthetic attitude 
 to the cognitive attitude towards one and the same material. To 
 skill and feeling is added knowledge. From using merely nominative 
 terms for the objects of sense we pass to systematic terms, which not 
 merely point them out on the basis of mental association, but which 
 indicate their place amongst other objects and their relations to them 
 on the basis of systematic knowledge. No objection can be brought 
 against this knowledge from the artistic or practical point of view, for 
 it builds upon the same ground as they do. It only adds the fullness of 
 knowledge to the sufficiency of sense. Then sense not only is present with 
 the mind and affects it to feeling and emotion, but it is known as well. 
 
 The primary tones of a chord blend with one another or with their 
 fundamental in the same general way as do harmonics. They are much 
 more easily recognised in the whole partly because they are louder, 
 partly because they are known and intended to be played. They are 
 part of the player's conscious intention, just as the blend or surface
 
 ix] AND OF ANALYSIS 61 
 
 of tone in its synthetic form is. We have already seen how tones an 
 octave apart may fuse so well together as to be mistaken for one. This 
 high fusion of loud tones approximates to that generally valid for the 
 weak tones of partials. 
 
 But the practised musician is able to pick out the primary tones 
 of a chord with considerable ease. The most gifted ear can pick them 
 out at once unfailingly in any part of the musical range and on any 
 instrument. This analysis cannot, of course, annul the underlying 
 synthesis of tones that is due to their volumic overlapping. But the 
 gifted ear can at once seize upon the pitch predominances that the 
 volumes contain, so as to cognise the component parts of the whole. 
 A clear analytic view is obtained without any of the synthetic effects 
 of overlapping or fusion being lost. Analysis, at its best in dealing 
 with primaries does not require the finest ear to pass successively 
 from one pitch-point to another in order to cognise them all. They 
 are all grasped at once, as any of us grasps the whole of a simpler visual 
 pattern or of a word in one gaze of fixation. 
 
 Nor is there here any general confusion between primaries and 
 harmonics. For the latter are heard on all familiar instruments as 
 synthetic blends, not as separate tones. If the grouping of tones makes 
 one or other harmonic very loud, this will tend to be heard as a primary 
 tone. But the prevailing attitude will be to distinguish only the loudest 
 components as primaries and to hear harmonics in their usual blend. 
 This attitude is greatly supported by the expectations made habitual 
 by the general course of musical spelling and grammar, into which 
 harmonics will not often fit coherently. 
 
 The less gifted ear does not distinguish the primary tones so readily. 
 It may well learn to recognise each interval and chord as a whole, as a 
 characteristic thing, as one learns to recognise words as a whole without 
 reading, or thinking of, each letter separately. And it may also then 
 readily learn to spell out the tones in the easier or more frequent 
 groupings, so as to be able at least to name the relative pitches of each. 
 But the first prevailing tendency is synthetic, even over and above the 
 inevitable synthesis of fusion; analysis of (stationary) chords is, then, 
 the result of effort and special attention. 
 
 But in melody the attention is almost relieved of any effort of analysis. 
 The analysis takes place as a matter of course ; or rather, the sequence 
 of tones that we call melody is purposely so formed that the attention 
 will follow it easily.
 
 62 THE RELATIVE IMPORTANCE OF SYNTHESIS [OH. 
 
 In primitive music there is commonly only one singing voice, which 
 displays some distinct melodic form whereby its movements acquire 
 unity and interest. In polyphonic music several voices proceed simul- 
 taneously, each one being melodically controlled in this way. In 
 harmonic music the fullest melodic treatment is given commonly only 
 to one voice, sometimes to two concurrently. The rest of the tonal 
 mass of each moment is handled synthetically, so that the listener 
 apprehends it rather as a whole. The sequence of tonal masses is 
 regulated partly by the requirements of melodic form and partly by 
 the relations connecting harmonic chords. Polyphonic music is, of 
 course, also a sequence of (harmonic) chords ; but the melodic treatment 
 of all the voices leads to a predominance of the melodic connexions 
 of the homologous voices of successive chords over the harmonic or 
 fusional connexions within each chord. 
 
 The degree to which the harmonic and melodic aspects of music 
 prevail over one another is thus very variable. At the one extreme 
 we find each voice so perfectly finished melodically that both the 
 artist and the auditor fail to apprehend the harmonic values of successive 
 chords in any special way, although they in no wise fail to hear how 
 far the different voices fit agreeably into one another's movements. 
 The basis of this agreeable conjunction is, of course, the fusional 
 relations of the tones of each chord. These are necessarily indestructible 
 and irremovable by any treatment of the attention or by any abstraction. 
 But nevertheless a special attitude of abstraction does lead the ear 
 to make as little as possible of them for the production of the larger 
 syntheses of the art. At the other extreme we find the melodic interest 
 completely subordinated to the harmonic. Each chord is a fusional 
 mass enjoyed for its special 'colour' or mass effect. The sequence of 
 chords is not decided on the basis of melodic form in any specific sense. 
 Melodic sequence, in general, is, of course, just as insuppressible and 
 irremovable as is harmony. The various chords that follow one another 
 must do so in such a way as to satisfy the minimal demands of melodic 
 movement generally. They must move by as small steps as possible 
 and must not cross one another, and so on. This minimum is enough 
 to guide the ear easily from one chord to the next, but it is not enough 
 to create any sort of melodic form. In fact the melodies that result may 
 be perfectly irregular. They only provide enough obvious movement to 
 guide the ear. Thus the mind is left free to devote itself to the harmonic 
 interests of the music, and the artist or improviser may pay his greatest 
 attention to building effects upon a synthesis of harmonic sequences.
 
 ix] AND OF ANALYSIS 63 
 
 The matter may be stated in a somewhat more figurative manner. 
 Each mass of sounds that constitutes music in several parts or voices 
 has two aspects : the one is its volumic aspect, the fusion characteristic 
 of it as a whole, or in any of its parts, i.e. between any two of its voices; 
 the other is its pitch aspect or its ordinal predominances, the points of 
 sound that stand forth intensely in it. The art that builds up chords 
 into complex music may, as it were, make either of these two aspects 
 the surface of the product that is to be exposed to the hearer, while the 
 other is made the mere surface of suture, cementing one brick of the 
 building to another. 
 
 If the ground of artistic synthesis is pitch, great care must be taken 
 in the selection of each brick that its pitch-points fit into those of the 
 next, so that the sequence will give a perfect complex of melodic figures, 
 easily surveyed by the listener. The subsidiary interests of the art 
 require the sequent chords to be so harmoniously consistent that they 
 will not severally fall to pieces or confuse the movements of the different 
 voices and will yet knit together so as to make a stable whole. But the 
 listener is not concerned with them beyond this. 
 
 If the ground of synthesis is harmony, that aspect of each chord 
 will be turned outwards. Sequences will be selected specially for the 
 manner in which they link together to form large harmonic surfaces 
 or masses, as it were. The melodic aspect is required only in so far as 
 it helps to bind the chords to one another on the unexposed surface 
 and so to perfect the underlying stability of the structure. 
 
 Or, if you like, in the one style of music harmony is put in the focus 
 of the listener's conscious mind, while melody remains in the background 
 of it; in the other style conversely. Or again, in the one harmony is 
 merely sensed and felt, while melody is built up into complex figures, 
 inspected, and watched in all its changes, and consciously enjoyed; 
 in the other the melody is merely sensed as an atmosphere, while the 
 specific artistic structure is harmonic. 
 
 All this comes to the same thing as John Hullah's oft repeated 
 dictum about the horizontal (melodic) and the perpendicular (harmonic) 
 views of musical structure 1 . The figure of speech is here derived from 
 
 1 v. 25, ice: "I use the word harmony as representing the successive results of an 
 accumulation of parts. For of a chord, as an isolated fact, the old masters took little account. 
 They were not harmonists at all, in our sense of the word, but contrapuntists; laying melody 
 upon melody, according to certain laws, but uncognisant of, or indifferent to, the effects 
 of their combinations as they successively came upon the ear. Their constructions were 
 horizontal, not perpendicular. They built in layers, but their music differs from most of 
 ours as a brick wall does from a colonnade," etc.
 
 64 SYNTHESIS AND ANALYSIS [CH.IX 
 
 the structure of the printed music in which the component tones of a 
 chord are written below or above one another, while melodies run from 
 left to right of the page through the tonal points of each chord. The 
 figure is, of course, not strictly applicable to what is heard. For the 
 horizontal aspect is not spatial, as the adjective suggests, but temporal; 
 the field of hearing if the pitch series (or the length of tonal volume) 
 be called its perpendicular dimension has no horizontal aspect at 
 all as far as music is concerned. But if allowance is made for this 
 discrepancy, the figure is apposite more so indeed than its originator 
 could have known. For the harmonic dimension is really akin to a 
 spatial dimension; it is ordinal, and space is probably an ordinal 
 derivative. 
 
 We have thus characterised in general the relative importance of 
 fusion and of analysis in music, and we have given these two aspects 
 of, or attitudes towards, tonal masses a basis in the nature of these 
 masses themselves as sounds. In other words we have shown upon 
 what features of tones fusion rests and what points of tonal volume 
 offer themselves for special analytic attention. In freely planned 
 experiments these attitudes may be prescribed, or imposed upon oneself 
 voluntarily and followed at leisure. As in other regions, so here it is 
 found that some circumstances make synthetic apprehension easy, 
 while others favour analysis or attention to a part rather than to the 
 whole. The work of the musical artist is to bring these two attitudes 
 under control, so that he may be able to guide the hearer's attention 
 to any aspect of tone he pleases; or so to construct his tonal masses 
 that listeners on the average will tend, with a minimal deviation, to 
 devote their minds to those aspects of tone upon which the artistic 
 effect has been built. For this purpose the artist must know as much 
 as possible which factors favour each attitude and what power each 
 factor has. Consequently the science of music is called upon to bring 
 these factors into the fullest light of knowledge and to explain as exactly 
 as may be how each one achieves its effect.
 
 CHAPTER X 
 
 THE EQUIVALENCE OF OCTAVES 
 
 THE equivalence of octaves at first glance seems clearly to rest primarily 
 upon the fact of the high degree of fusion appertaining to coincident 
 tones an octave apart. For that is the ultimate fact of tonal hearing 
 that most resembles the equivalence of octaves in music. Such pairs 
 are very often mistaken for a single tone, more often than happens 
 with any other interval. When octave tones are sounded in succession, 
 there is of course no such approximation to the sound of a single tone, 
 but there is an evident connexion between the two which reminds us 
 of the transition from a thing to its replica, and which we therefore 
 incline to call by the name of similarity or identity or equivalence or 
 the like. Which of these terms is used, depends apparently upon the 
 relative importance either for theoretical or for practical purposes 
 that is ascribed to the sameness and to the difference of the two tones. 
 For octave-tones are obviously not absolutely the same. 
 
 But although the primary basis of the equivalence seems so obvious, 
 the system of facts of a similar nature does not seem to confirm it, 
 at least in practical connexions. For the octave is only the highest 
 degree of a series of grades of fusion which have been known more or 
 less satisfactorily since the earliest days of the science of music. This 
 series would lead us to expect a similar grading of equivalence, which 
 by no manner of means can be claimed as real. We cannot call the 
 tones of a fifth similar or equivalent as we call those of the octave, 
 not even if we say the degree of similarity or equivalence is very much 
 less than in the octave. It is true that crude and primitive forms of 
 music do use parallels of fifths in the same way as we use parallels of 
 octaves in our music. But even so the use is nothing like so extended, 
 nor has it survived the first refinements of musical taste. Fifths are 
 then no more equivalent than fourths or thirds or seconds are. But 
 the equivalence of octaves is of the greatest and most extended 
 importance in all music; far from being merely a primitive crudity, 
 it increases in importance with the development of music. 
 
 Its central importance for musical practice and theory dates from 
 the famous doctrine of Jean Philippe Rameau concerning the inversions 
 
 w. F. M. 5
 
 66 THE EQUIVALENCE OF OCTAVES [CH. 
 
 of chords. In the preface to his simplification of Rameau's teaching 
 D'Alembert pointed out that up till then work "had been confined 
 almost completely to the collection of rules without reasons for them; 
 there had been no discovery of analogy and of a common source; 
 blind trial had been the sole compass of artists." "M. Rameau," he 
 wrote, "is the first to begin to dispel this fog of chaos. In the resonance 
 of the sonorous body he has found the most probable origin of harmony 
 and of the pleasure it causes us : he has developed this principle and 
 shown how the phenomena of music emerge from it : he has reduced 
 all the chords to a small number of simple and fundamental chords, 
 of which the others are only combinations and inversions; finally he 
 has succeeded in apperceiving the mutual dependence of melody and 
 harmony and in making it felt" (9, vi f.). "Whatever may be the 
 fruit of the further efforts of others, the fame of the learned artist has 
 nothing to fear; he will always have the merit of having been the 
 first to make music a science worthy to occupy philosophers; of having 
 simplified and facilitated its practice; of having taught musicians to 
 carry into this region the torch of reasoning and of analogy" (9, xviii). 
 Later on (9, 222) he wrote that a certain special difficulty and some 
 others less considerable, would not prevent fundamental basses from 
 being "the principle of harmony and of melody; as the system of 
 gravitation is the principle of physical astronomy, although this system 
 does not account for all the phenomena that are observed in the move- 
 ment of the celestial bodies." The idea of the connexion between 
 chords that involve the same notes of the octave is so familiar to the 
 modern musical mind that it is necessary to recall clearly that the 
 idea did not always stand in the forefront of the musician's cognitive 
 consciousness. He may always have felt it, to be sure, but he certainly 
 did not always know that he felt it 1 (cf. 60, 43). 
 
 1 Readers who look upon the connexion of inversions as perfectly obvious will be 
 interested in a quotation from a contemporary of Rameau's to whom the latter's doctrine 
 came as a novelty : 
 
 "We must not omit an observation most easy to make at this point and also of the 
 greatest moment for the clearness and solidity of the doctrine we have been gradually 
 expounding. A concert has need for example of three voices if it is to embrace with their 
 help three consonances, prime, third and fifth, which are called by the masters Harmonic 
 Triad. The prime is always found placed in the lowest, the third in the middle, the fifth 
 in the highest place. Now suppose that the prime is moved to the higher octave, so that 
 the third remains in the lowest place. The ear is no longer satisfied with it. It no longer 
 seems that the concert is finished. Hence the concord does not feel that it has yet returned 
 to that note whence it has taken, and in which it recognises, its origin and in which alone 
 it can come to rest and finish. It is openly apparent that the harmony is suspended.
 
 x] THE EQUIVALENCE OF OCTAVES 67 
 
 Rameau himself actually thought that we fail to distinguish octaves 
 in "the resonance of the sonorous body." The partials 1, 2, 4, 8 and 
 16 are, of course, octaves, which, he said, really resonate even more 
 loudly than do those numbered 3 and 5, because of the size of the 
 resonating parts of the musical instrument. So even though we fail 
 to distinguish them, we are nevertheless necessarily affected by them, 
 "but by an occult feeling that has so far prevented us from discovering 
 its cause" (54, 3). From this feeling our sense of the identity of octaves 
 has arisen. We actually prefer to have tones closer together than they 
 are offered to us by nature in the series of partials, in order that we may 
 have them within the range of the voice. For as we do not distinguish 
 the octaves amongst the partials, the range of the voice is soon exceeded. 
 Thus, 1 , 3 and 5 take us up through a range of two octaves and a third, 
 and they actually include only one interval less than the octave, namely 
 the major sixth between 3 and 5. The ear also finds it easier to move 
 about amongst close intervals because of the short distance between their 
 tones. But the identity of octaves does not prevent them from intro- 
 
 This prime voice has nevertheless not been omitted. We have done naught but transfer 
 it from the lowest place, where it stood, to the highest. We have still in ear the same three 
 notes. How then has so great a change in the effect of all been made? To imagine hearing 
 the low octave of a high note that we actually hear is very easy for anyone. So in this 
 way we shall be able to make up for the defect, replacing in fantasy the true bass in the 
 place whence it was taken. Then we shall have this principal voice present with us in 
 two places: once in the high part where we hear it, and again in the low part where we 
 imagine it. Our ear will nevertheless not yet be satisfied. We shall still not hear the perfect 
 chord, the chord that concludes. And why so ? Because the force of the high voice that 
 is really heard prevails over the force of the low voice that would only be imagined. 
 The sense of the ear that is the natural judge of harmony, does not let itself be deceived 
 by the imagination. It would still refer the two prime voices that are actually heard to 
 the third which is also sensed; and thus the harmony would still remain imperfect. It 
 would not refer it to the imaginary fourth voice by reference to which alone the two 
 higher notes could change their proportions and render themselves apt to conclude. 
 This most simple observation which turns upon an experience known to everyone and 
 beyond all doubt, proves that the common statement that the one octave is the equivalent 
 of the other requires some limitation. In a large number of cases the statement is true, 
 but not in all. In particular it is always false in reference to notes that do duty as bass; 
 the which in changing place change their nature and make the nature change of all others 
 from below which they withdraw. 
 
 Now if that is so (and nobody can deny it), however could truth or at least verisimilitude 
 belong to the new doctrine of inversions which is nowadays so celebrated as a thing 
 most useful to the art and perhaps the most noble secret that has yet been discovered in 
 harmony ? 
 
 To me there seems to be nothing to recognise in it but error and perversion" (57, ssff.). 
 
 It is clear from the above that the 'sameness' of octaves is not the same idea as the 
 equivalence of inversions (cf. 60, w). 
 
 62
 
 [CH. 
 
 ducing some differences into harmony and melody. But that, Rameau 
 said, consists "only in the different modifications of one and the same 
 whole differently combined, where sounds cannot change their order 
 without the help of their octaves" (54, 13). The sounding of an octave 
 in place of the fundamental in no way distracts the ear from the natural 
 whole that guides it; the ear recognises the fundamental sound in its 
 octaves, no matter what the order of the parts of the chord ; it is always 
 reminded of this same whole (given in the resonance of the sonorous 
 body). If the chord is consonant, it is equally so in all its combinations. 
 In short, 2, 4, 8 and 1 are for us but one sound, in which 1 always 
 presides, whether we hear it or not (54, 16). Identity, Rameau added, 
 may seem rather an extreme term to use, but you may adopt any 
 term you like so long as, not going so far, it goes far enough. 
 
 That is precisely the difficulty in this problem to find a theoretical 
 basis that will evidently go as far as is needful in establishing sameness 
 and with equal evidence refrain from obliterating the differences that 
 feeling and practice demonstrate. Rameau certainly overdid the aspect 
 of sameness. He admitted himself that we follow in our music the 
 traces given by nature in the resonance of the sonorous body "only 
 by the grace of these octaves" (41, 36). That is perfectly plain; it has 
 often been laid as a primary difficulty against those who claim to derive 
 the tones of the scales from the series of partials. How are you going 
 to bring them down from their heights to within the range of an octave? 
 Some second principle is obviously required for this purpose. This 
 was given for Rameau in his "occult feeling." Without that the needs 
 of the voice would remain unsatisfied; or rather the voice would have 
 had to be devised so as to cover a much larger range. And the ear 
 would likewise have had no scope for preferences as to the sizes of 
 intervals. 
 
 Rameau was also right in claiming that in the different inversions 
 we are reminded of a certain whole, but that whole is not the octaves 
 1, 2, 4, 8 and 16. Such an answer would be elicited from the mind of 
 no musician unlearned in the claims of theory. But any musician would 
 answer more or less so that the whole recalled to his mind is the 
 group of all possible combinations of the chord. When asked to name 
 a given one of them, he will say : "that is (the arrangement of) the 
 inversion of the chord." When one of these combinations is heard, 
 the common relations are also 'in some way' heard, but the differences 
 peculiar to it are equally evident. Though the parts out of which the
 
 x] THE EQUIVALENCE OF OCTAVES 69 
 
 chord is composed and 'in some manner' the whole that it forms are 
 always the same, yet the consonance is by no means the same, nor is 
 the harmonic treatment, although it may well be the case that inversion 
 will not turn any consonance into a dissonance, or conversely. For the 
 purposes of science, it is just the 'some way' and 'some manner' that 
 is the problem. That is what we must give precise form to, so that the 
 practical and sensory consequences so clear to the musician may follow 
 evidently from as clear a conceptual foundation. 
 
 Helmholtz's theory has seemed to many to be a great improvement 
 upon Rameau's or even to have finally solved the problem. In the 
 octave we hear again a part at best the half of what we heard 
 before the fundamental and its own special series of partials. A most 
 suggestive and winning explanation, very hard to abandon even when 
 it has been disproved on other grounds! A theory is always seductive 
 when it has a fair and clear speech for every phase of the business, for 
 every doubt and hesitation, and withal so cleverly conceals the fact 
 that the basis of explanation has only been assumed; this basis is not 
 really patent and clear, as it is in the analogous cases referred to for 
 support the synthetic similarity of faces; it is only 'just as good.' 
 What Helmholtz failed to show was why partial tones ever come to 
 form a fused synthetic whole. And it is difficult for most folks to appre- 
 ciate the importance of this omission. The explanations which flow 
 from the assumptions are for all ordinary cases apparently so neat and 
 apt that more could hardly be desired. Further demands and criticism 
 look like finical pedantry. 
 
 And yet Helmholtz secured his whole basis of explanation by mere 
 analogy one of the kind that can be stated so plausibly for either of 
 two opposite ends. For if in the octave we hear again a part of what 
 we heard before, that should lead us in the course of time to distinguish 
 the first and second partials of a tone as different primary tones. The 
 progress of musical practice would thus bring about a gradual analysis 
 of timbre into its ultimate constituents. Some psychologists believe 
 that the world begins for the child in William James's words, as "one 
 great blooming buzzing confusion" (27, 488). But we know that it soon 
 clears up into its many distinct parts. Why should it not be so with 
 the parts of musical tones or of ordinary tones? Separation and separate 
 handling should here also lead to mental distinction and abstraction. 
 Who is to hold the balance between the tendency to confuse the parts 
 of a whole with one another or with the whole and the tendency to
 
 70 THE EQUIVALENCE OF OCTAVES [CH. 
 
 distinguish the separable parts of the whole from one another or from 
 the whole? Helmholtz, after all, gets no further than does Rameau 
 with this "occult feeling." In fact this phrase better conforms to the 
 results of Stumpf's criticism of the theories of consonance given by 
 Helmholtz and others and to the suggestions finally made by Stumpf 
 as to the probable basis of fusion in some synergy of the nervous 
 system. The " occult feeling that has so far prevented us from discovering 
 its cause" is just what we might expect from 'synergy,' which is an 
 occult (cerebral) process that has so far prevented us from formulating 
 its nature. 
 
 In recent years an attempt has been made to account for the equi- 
 valence of octaves by setting up the series of differences that lie within 
 the range of an octave, no matter what its general pitch may be, as 
 the primary qualities of hearing. Then the series from c to c', whether 
 it be taken continuously or discretely as in any specific scale, is a series 
 of qualities like that of the spectral colours; only in the tones we do 
 not merely just return to the starting-point but we are able to repeat 
 the series a number of times. 
 
 This theory can hardly be discussed without close study of the 
 psychological notion of quality. That includes all kinds of sensation 
 such as touch, cold, warmth, pain, sweet, sour, the various smells, the 
 colours such as blue, red, etc., muscular feeling, hunger, thirst, etc., etc., 
 all as specific feelings, without concern for their intensity or localisation 
 or for any other distinguishable aspects of them, except merely their 
 kind. This bare kind or quality is the thing of all things that we know 
 perhaps least of in itself. We seem to have some understanding of it 
 in vision; but our understanding is here almost solely physiological. 
 None of us knows what inner connexion, if any, there is between blue 
 and red, or between yellow and blue, as felt colours. There seems to 
 be none, and yet we at once recognise them all as colours. 
 
 One of the characteristic features of colours is their changes of kind. 
 Red passes through orange, that resembles it, to yellow, that is like 
 orange, but not at all like red. A similar change brings us to green, 
 then to blue, and finally back to red through purple. If we are to consider 
 the differences included within the octave as qualitative, comparison 
 with colour would incline us to look for characteristic turning points, 
 as it were, within the octave. These might be supposed to occur at 
 the thirds, fourth, fifth and sixths perhaps. Various suggestions have 
 been made. But none of them really explains the peculiarities that
 
 x] THE EQUIVALENCE OF OCTAVES 71 
 
 would be thus described 1 . It is, of v course, conceivable that from a 
 detailed study of classifications made in relation to the various forms 
 of tone-deafness, etc., a good and probable physiological theory of 
 tonal quality might in time be obtained, just as has been done in vision. 
 Such a theory might then explain the peculiar relations that characterise 
 thirds, fourth, fifth, and the rest. No very satisfactory explanation 
 has as yet, however, been given even of colour affinities. The prospects 
 of raising a lucid theory of music on this qualitative basis are, to say 
 the least, not yet exciting. 
 
 Of course, that would be of no consequence at all if the classification 
 as quality were logically inevitable. It is not so by any means. Some 
 objections may be raised to the theory from the special difficulties it 
 creates, from the obscurity of the ground it rests upon, and from the 
 special phenomena of quality which the classification must introduce 
 (cf. 77, 44 ff.). But until exclusion makes one theory or another logically 
 inevitable, the merits of theories rest upon their respective powers of 
 accounting for all the facts. The theory of octave qualities does not 
 reduce the amount to be explained. The assumptions it makes require 
 as much explanation and justification as do the facts they are supposed 
 to explain. And a better explanation can be given without them. 
 
 The intervals in common use and their inversions are reducible 
 to six pairs: 
 
 9 TT 1 TTT 4 T 
 
 -i, AJ., O, -HI, -*, 
 
 VII, 7, VI, 6, 5, T. 
 
 The only marked change in grade of consonance produced by inversion 
 is found in 4 5 4. The fifth is clearly more consonant or fused than 
 the fourth. But in all but the tritone a decided difference is wrought 
 in the interval itself. In the volumic theory of tone already developed 
 
 1 As the octave according to the volumic theory is the greatest approximation towards 
 the balance of a single tone that two simultaneous tones can make, and the fifth is the 
 next, we might expect a certain parallelism in the character of the steps by which we 
 pass from the two ends of the octave to the fifth : 
 
 o, VII, 7, VI, 6, eX-.. 
 
 p, 2, II, 3, III, 4 / Tnt0ne 
 
 Here the fourth is taken as the counterpart of the fifth, as it were. Otherwise the parallel 
 will only hold if the fourth is slumped with the thirds and the tritone is set over against 
 the minor sixth when it functions (in equal temperament) as a discord (augmented fifth). 
 Thus: 
 
 o, VII, 7, VI, 6, 6 5 , 5 
 
 p, 2, II, 3, HI, 4, T, 5. 
 Cf. Chap, xxi below.
 
 72 THE EQUIVALENCE OF OCTAVES [CH. 
 
 we have good ground for the understanding of both these facts. The 
 balance and symmetry of very different intervals may be approximately 
 equal. But the intervals themselves are so different because they are 
 quite different proportions of volumes. 
 
 Now the musical ear is not restricted to a knowledge cf the simplest 
 intervals. These are naturally of great importance in music, because 
 they are amongst the simplest complexes of form known to the art. 
 The simplest of all is the absolutely pure tone. A variant upon this is 
 the blend, which gives the tone a surface, as it were. Interval is the 
 first step that involves a variable proportion, constant only for each 
 specific interval. But it is only the first step on a long line of possible 
 complications, each of which may equally well be learnt as a definite 
 complex of proportions, or as a 'pattern.' Let us follow out this process 
 of complication, beginning with the addition to a simple interval of 
 the octave of its lower tone. The 'chord' c, e, c 1 , for example, may be 
 represented thus (Fig. 3) : 
 
 Fig. 3 
 
 The length of the lines represents the relative length of the volume 
 of each tone and the middle points mark their pitches the points 
 that predominate in each volume and so give the whole a definite 
 mark by which it can be placed in the series of all the tones. The volume 
 of a tone is, of course, not homogeneous throughout its length, as the 
 simple line suggests. It probably varies from the central pitch-point 
 towards either end by a regular decrease of intensity. In any case this 
 variation is quite regular, so that each tone may be a symmetrical whole. 
 The range of the variation from the pitch maximum to the opposite 
 ends of the volume's length will probably be the greater, the louder 
 the tone is. Consequently when several tones overlap to form a chord, 
 the intensity at each part of the chord's volume will vary infinitely 
 according to the relative strength of the component tones. For the 
 overlapping will give some sort of summation of the strength of each 
 particle of sound that is common to two or more of the component 
 tones of the chord. We cannot yet say precisely what the mode of this 
 summation is. However, there is a feature of every chord that is in 
 no way affected thereby, namely the relative or proportional position 
 in the whole volume of the points where a departure from the regular
 
 x] THE EQUIVALENCE OF OCTAVES 73 
 
 changes that constitute the balance and regularity of a single tone 
 occurs. And these points are bound into definite sets by their dependence 
 on the physical stimulus of sound. They are always the same for any 
 one ratio of vibration. And they are psychically fixed by the ordinal 
 character of the points themselves. Thus the whole volume will be 
 marked out into a set of proportional parts properly indicated in the 
 diagram given above. 
 
 The musician in the course of his practice is made thoroughly 
 familiar with the complex cec 1 both as a whole and in its parts, c, e 
 and c 1 , and their binary combinations, ce, ec 1 , cc l . In time he becomes 
 able to survey these parts within the whole and to recognise their 
 presence, either in an absolute way by naming their exact pitches, or 
 in a relative proportional way by recognising the intervals they form. 
 Even though he cannot banish c from the whole complex, he can survey 
 the 'upper' parts around the pitch-points of c 1 and e and recognise the 
 proportions of these. Or he may think of, and attend to, ec 1 and recognise 
 its presence, ignoring e the while. Or he may dwell upon ce and ignore c 1 
 or at least its predominant parts about its pitch-point. This process 
 of abstraction is already familiar in all those other senses which show 
 a definite field or an ordinal system, such as touch and vision. We can 
 shift the attention easily from finger to finger so long as touch sensations 
 appear in either. In vision we are much more expert at such spatial 
 or ordinal abstraction. The patterns of wall-paper often allow of com- 
 bination and recombination in the most varied way. 
 
 The special peculiarity of hearing in this respect is the relative 
 slowness with which the average person acquires practice and skill in 
 recognising tonal proportions and in abstracting them from complexes 
 of tones. In the visual field we can move patterns from place to place 
 or rotate them, and dissect them as we please. In hearing rotation is 
 impossible; movement within the sensory field is only possible in so 
 far as pitch and volume of tone are altered in the way laid down by 
 the physical stimulus; and dissection is limited in the same way. These 
 restrictions make analysis so hard that most people are discouraged 
 by them. But they are easily enough overcome by those in whom a 
 good ear has created special interest and enthusiasm. 
 
 As to the way in which we may judge of the fusion of parts in the 
 whole by abstraction from the whole, there is some difference of opinion. 
 It has been urged that fusion the degree to which a tonal mass appears 
 to resemble the unity of a single tone must necessarily be, and is,
 
 74 THE EQUIVALENCE OF OCTAVES [CH. 
 
 modified by the addition of a third tone to any pair. The mode of 
 alteration will depend on whether the new tone forms a greater or 
 less fusion with either of the two tones than they form with one another. 
 Yet one might have expected the united fusion always to be worse, 
 since the new tone necessarily forms a lesser degree of unity with either 
 of the first two than that one formed by itself, being a tone, i.e. an optimal 
 unity. Therefore when this deteriorated tone is added to the other 
 one of the pair first given, the triad resulting should always be less 
 fused than the original pair. Probably a good deal depends upon the 
 point of view. If we look for mere plurality of tones, any trio will be 
 more plural than the duo. If we look for the amount of good balance 
 or fusion, as that is known in the octave, fifth, etc., we shall find more 
 of it present in cgb than in cb. Here mere interpenetration over the 
 whole tonal mass is not so much the standard, as perhaps a certain 
 kind of interpenetration already familiar in various forms. There may 
 be some abstraction in the process i.e., a local abstraction within the 
 ordinal field of sound of the chords. It is evidently not easy for those 
 who experiment upon the fusion of more than two tones to make their 
 point of view in observation quite clear. The opposite view has been 
 upheld that the addition of further tones makes no difference whatever 
 to a fusion already given. This seems quite a reasonable position 
 provided the above-mentioned 'local' abstraction of fusion has become 
 easy enough. 
 
 There seems to be no reason in the nature of tonal complexes them- 
 selves why such abstraction should not succeed with those who are 
 highly gifted and practised acoustically. They would then isolate for 
 attention the tones in question and see the sort of balance and symmetry 
 they possess. Of course they cannot lift the tones they abstract out of 
 the whole complex they are abstracted from. Abstraction here means 
 only devoting special attention to certain tones and recognising in them 
 features that are usually characteristic of them in isolation, in so far 
 as these features have been only partially or not essentially distorted 
 by the presence of the other tones. Thus one who abstracts cc l from 
 the chord cec 1 will notice that the maximal volume of the chord is that 
 of c; that c 1 is present as a pitch at its usual ordinal place, that the 
 parts of the tone lying around the pitch-point have the proper tonal 
 symmetry and that there is the usual clean function at the pitch-point 
 of c. Of course the tone e will often be encountered during this process. 
 But one who is highly practised may pass to and fro about this irrelevant 
 tone without being disconcerted by it, and may feel as able to give
 
 x] THE EQUIVALENCE OF OCTAVES 75 
 
 his judgment as he would if it were not there. For others, however, 
 the third tone may be a source of great disturbance and they may feel 
 they never really can ignore it, so that it always spoils the effects for 
 them. 
 
 Later on we shall meet evidence that will call for a more special 
 effort to settle the question of the part played by the fusion of single 
 intervals in chords. 
 
 The ' chord ' cec 1 is in the experience of the musician not only given 
 at all levels of pitch, but when it occurs at any pitch, it then commonly 
 occurs at the octaves above and below. Thus cec 1 may be carried up 
 and down over the piano, cec 1 e 1 c 2 e 2 c 3 (Fig. 4). The new parts here make 
 no significant change in the diagram of volumes, e 1 fits in between the 
 pitch of e and the common upper limiting point of all tones, while c 2 
 
 Fig. 4. Illustrating the continuity of 'pattern' made possible by the volumic relations of 
 the octave and upon which the connexions of inversions of the 'same' chord rest. 
 
 likewise fits in between the pitch of c 1 and that point. Both of these 
 latter tones can be taken as mere appendages of their lower octaves, 
 the more go the weaker their strength is. In any case the new tones do 
 not spoil the previous pattern, but merely continue it further towards 
 the upper limit of hearing in the same characteristically proportionate 
 form. The component tones of the whole are not more easily separable 
 for their being so many but less so. Only, the characteristic pattern 
 of the whole remains the same and can be recognised with almost, if 
 not quite, equal ease. 
 
 Now if ec l e l or its extensions are given in the same way, they may 
 not only be analysed into the same musical components c and e, but 
 they give a pattern which is partly the same as that of cec 1 . The closeness 
 of the resemblance is the greater, of course, the further up the pattern 
 is extended, ec 1 e 1 c 2 e 2 , etc. But it is obvious that cec 1 and ec l e l are 
 identical in respect of their common part ec l . This part will make 
 them thus far similar. And the resemblance is increased by the similar 
 way in which the other tone is related to one of the tones of the common 
 pair. Of course this sort of similarity is evident both in the mere musical 
 symbols and their arrangement and in the common musical consciousness
 
 76 THE EQUIVALENCE OF OCTAVES [OH. 
 
 of our time. Our concern here is to show definitely how this similarity 
 is grounded upon the sensory material of hearing, in the tones themselves. 
 The similarity that appears to the musical mind is therefore not merely 
 the result of musical analysis or of theory or of thought, but is a true 
 representation of the relations of the parts of the sensory stuff of music. 
 
 Thus we do right to consider cec 1 and ec l e l in a certain respect as 
 mere aspects of one another, or to consider ec^e 1 as a trifling alteration 
 of cec 1 , which for some reason we look upon as the normal or more 
 fundamental form. The same holds for any other chord, no matter 
 how complex and discordant, ceg in a certain respect appears again 
 in ego 1 and in gcP-e 1 . They are all patterns that may be said to be parts 
 of their common extension cegc^g 1 ^, etc., except that the common 
 pattern does not begin at the same part of its cycle, so to speak. 
 
 The musical consciousness that has got thus far, will find it easy 
 to see the same pattern even when its parts are scattered more widely 
 through the octaves of its extension. Thus we come to forms such as 
 cge 1 , ce 1 */ 2 , ge l c?, the familiar positions of the various inversions. The 
 connexion of these with the fundamental pattern ceg cannot remain 
 obscure after the musical mind has learned so much as to be able to 
 create ceg itself, to know it and to use it. Of course this pattern is only 
 relatively feebly indicated in cg l e 2 ; but for the practised musical mind 
 and we are here dealing only with practice in the simplest things, 
 though for the foundations of the science they are the hardest problems ; 
 the connexion is as plain as daylight, as plain as is the ordinary hand- 
 writing of our own language in spite of its great variations from the 
 copperplate model. 
 
 In ordinary music, moreover, there is a much greater resemblance 
 between cg l e 2 and ceg than appears in the great intervals between the 
 parts of the former, or in the diagrammatic representation of it on the 
 basis of absolutely pure component tones. For these fundamental 
 tones are ordinarily accompanied by upper partials. If we suppose 
 merely that the lower partials are present in some strength, we should 
 get 
 
 c 
 
 c c 1 g 1 
 
 c 2 
 
 e 2 
 
 g 2 (* 
 
 t>) c 3 (d 3 ) 
 
 e 3 g 3 
 
 
 
 g 1 
 
 9 1 
 
 
 
 g 2 
 
 (d 2 ) 
 
 g 3 
 
 (b 3 ) (c 
 
 I 4 ) g 4 
 
 e 2 
 
 
 
 e 2 
 
 
 
 e 3 
 
 (b 3 ) 
 
 e 4 g*3 
 
 Total 
 
 c c 1 g 1 
 
 c 2 
 
 e 2 
 
 g 2 
 
 c 3 
 
 e 3 g 3 
 
 
 e* g 4 
 
 In this series the pattern ceg is represented more than twice. The 
 resemblance would therefore, be more evident in instrumental tones
 
 x] THE EQUIVALENCE OF OCTAVES 77 
 
 than in pure tones. The musician here is not expected to analyse 
 partials a thing he rarely does at all. The point is only that these 
 partials will reinforce an effect that is apparent enough to him already 
 on common psychological grounds without partials. Partials alone 
 would not suffice, without the volumic basis. But granted that, they 
 will only repeat and confirm it. The same holds true to some extent 
 for the difference-tones. Thus between tones whose vibrations stand 
 in the ratio of 1 : 3 (e.g. c and g 1 ), the first difference-tone (h-l) will 
 have the ratio 2, and so will form even in the case of pure tones a 
 link towards the filling out of a pattern and the extension of connexions 
 by fusion beyond the octave. 
 
 But in all this we must not omit to notice that there are marked 
 differences between the different inversions and their different positions. 
 Therefore we observed above that these forms were identical 'in a 
 certain respect.' Apart from that and for other purposes, their differences 
 are great; and naturally too. The bass is, as above explained, the 
 weightiest part of the chord, its centre of gravity so to speak; and it 
 must make a great difference which part of the basal pattern bears 
 this function. That is perfectly familiar in musical practice and theory, 
 and just as clear on our theory. The pattern set by g&e 1 continues as 
 g l c 2 e 2 , etc. It has the same series as ceg, once it is well started; but, 
 as given, it designates the g^e 1 complex unit of pattern most strongly. 
 If we care to ignore that designation or are specially led to do so, then 
 we may well see the ceg type most of all. What the special differences 
 between these types of the same basal pattern, as it were, consist in 
 essentially, we shall endeavour to show as we proceed. 
 
 We have thus shown that the equivalence of octaves rests upon a 
 sufficient natural basis in the stuff of tones themselves. And our account 
 in no way inclines us to underrate the differences that exist in that 
 sensory stuff between the different groupings of tones that are equi- 
 valent. There is only equivalence for certain purposes. A point of 
 view, an attitude, or a certain trend of abstraction has to be made 
 for the equivalence to emerge so strongly as to suggest sameness. 
 Other attitudes may concentrate in other ways, and see practically 
 nothing but difference. For certain purposes there is familiarly a very 
 considerable difference between ceg or egc 1 and gc l e l . 
 
 The equivalence thus established for octaves does not apply to any 
 other interval in the same way. Suppose we double the interval of the
 
 78 THE EQUIVALENCE OF OCTAVES [CH. 
 
 fifth cgd 1 (Fig. 5). The second fifth does not fit into the first so as 
 to be a mere repetition of its pattern. It would, no doubt, do so, if 
 each tone consisted only of the half of its actual volume that lies on 
 the upper side of its pitch (u in the diagram and not I). But these I 
 parts break into one another irregularly. The Z-end of the second fifth 
 strikes in between the pitch-points of the two lower fifths. In the case 
 of the octave the I parts of the higher tones merely repeat or emphasise 
 the pitch-points of their lower octaves, so that no new or disturbing 
 element is introduced. The higher tones merely carry onwards and 
 upwards the pattern already given by the simple interval or chord. 
 Of course we can attend to the one or other fifth in the whole and 
 hear it as a fusion, but the two do not follow upon, or fit into, one 
 another as a continuation of one whole pattern. If gd 1 is played after 
 eg, and the attention is concerned with their justness as fifths, gd 1 
 will be heard as the repetition of eg. Or if the attention is concerned 
 with the indirect relation holding between c and d 1 through a real or 
 imaginary g, it will take a similar attitude. But if gd 1 follows eg as a 
 
 I 
 
 Fig. 5 
 
 part of a whole to which both belong, the attention directed to this 
 pattern will not find itself rewarded. This shows more clearly than 
 does the octave that equivalence of octaves is not the mere repetition 
 of a tone of the same 'quality' absolutely inherent in itself, or of the 
 same single interval of two tones; but it is the presence of a pattern 
 of which a chord of any number of tones forms a part, or the indication 
 of that pattern in a way that suffices for the musical ear under the 
 circumstances of the moment. 
 
 As the fifth is thus distinguished from the octave, so are the ether 
 intervals. The octave is the only interval that thus makes possible 
 the extension of patterns. And it does so because it packs the repetition 
 entirely into the upper half of its lower tone, making only one new point 
 of predominance at the upper pitch-point. That is why the octave 
 is of such fundamental regulative importance in all music, and why 
 the equivalence of the octave, instead of being a survival from primitive 
 forms of music, is of constantly increasing importance. 
 
 The explanation we have given also shows that the equivalence 
 of octaves does not rest entirely on their great fusion, as such. Of course
 
 X] THE EQUIVALENCE OF OCTAVES 79 
 
 the reason for the great fusion is closely allied to that which makes 
 equivalence possible. But the equivalence does not rest on the balance 
 of proportions. For if it did, the fifth as already noticed, would provide 
 an equivalence of second grade, which is not really found. The fifth 
 holds a steady second place in music to the octave only as a fusion, 
 i.e. as a consonance in our music, or as a form of homophony in primitive 
 music. Equivalence rests, not upon the balance of the parts of a single 
 interval, but upon the way in which octaves extend the pattern of 
 proportions of an interval or of a chord without distorting that pattern. 
 Equivalence thus introduces an important new form into musical 
 structure. This form is present, indeed, in mice in the single interval, 
 but it only emerges clearly as an important specialty when chords are 
 freely used, and when they have been for some time steadily apprehended 
 for the purposes of musical structure in a special way, i.e. not as fusions, 
 but in another way, which we have classified as 'pattern.' Thus we 
 can now well understand why the notion of the equivalence of inversions 
 did not take clear shape in the musical mind until the period of harmonic 
 music had been fully inaugurated and had had time to ripen into 
 conscious formulation in Rameau or his more immediate precursors. 
 
 In other words that make it almost a truism, a system of connexions 
 like those of inversion is only possible when the terms connected have 
 become familiar. Simple though this is, it is so important that it may 
 be set up almost as a principle for the study of chords in so far as the 
 notion of inversion reduces these to a manageable number. We can 
 speak of inversion properly only when we know that the best (average) 
 listeners are so familiar with the different chords as to be able to recognise 
 them readily as parts of the same pattern. After all any set of notes 
 whatever can by suitable (octaval) transposition be inverted into a 
 series of major or minor thirds with appropriate omissions. No real 
 system of chords can be founded on such merely formal considerations. 
 The primary factual question for every system of chords that uses the 
 notion of inversion is : are the inversions recognised by direct hearing 
 as parts of one pattern ? Contrariwise, the formal reduction of all chords 
 to columns of thirds does nothing at all to prove either the real 
 importance or the real primacy of the third in musical structure. 
 
 The preceding exposition cannot, of course, prejudice the efforts 
 of abstraction in listening to chords that may still become possible 
 to the musical mind. It is conceivable, for example, that a mind might 
 contrive to attend to a column of simultaneous fifths or of any other
 
 80 THE EQUIVALENCE OF OCTAVES [CH. x 
 
 interval apart from their mutual interference and blurring. Great 
 concentration and practice in distinguishing pitches rather than 
 intervals might lead this way. Those who have absolute ear often 
 recognise intervals rather by inference from their absolute pitches 
 than by direct apprehension of intervallic proportion. So a mind 
 might come to hear chords as columns of pitches standing at propor- 
 tionate distances from one another rather than as volumic patterns 
 of proportional nature throughout. Of course much that is of the 
 greatest value, if not essential, to music, would thereby be abandoned- 
 all harmonic effect in particular. Perhaps some of the latest experiments 
 in music-making tend in this direction; for example Scriabin's columns 
 of 'fourths.' Only the further developments of this line of construction 
 and the general judgment passed upon it in the course of time will show 
 whether it has struck upon new and useful faculties of musical analysis 
 that will serve the synthetic ends of artistic creation. If we cannot 
 discover whether Scriabin had a special attitude of listening to his 
 own music, we shall have to see whether in time such an attitude will 
 not prove to be essential for the artistic apprehension of his works.
 
 CHAPTER XI 
 
 CONSECUTIVE FIFTHS 
 
 THE rule forbidding consecutive fifths is one of the fundamental 
 generalisations of musical structure. The view is indeed sometimes 
 expressed that modern developments have swept all the rules of harmony 
 away, and that this one like others no longer holds because composers 
 break it repeatedly. It is true that the rule has its exceptions. But 
 the special means required to make such exceptions tolerable and their 
 late appearance in any frequency in the highly developed art show that 
 the rule has really the fundamental importance commonly ascribed 
 to it. The breaking of an established rule is naturally the first fact 
 to engage the attention, when it has been broken. The next question 
 inevitable for a mind that feels the good effect produced in spite of the 
 breach of rule is : what other elements of the whole in which the fifths 
 appear, are responsible for the good effect? The pleasantness of fifths 
 in a certain setting by no means discredits their prohibition under 
 most circumstances. This could be gainsaid only by the pedant who 
 lives on rules and does not apprehend the structures he studies in their 
 primary aspect aesthetically at all. But the sole standard of art 
 is the beauty inherent in the created object. We do right to expect 
 art to be, like nature, a realm of law and order, not the sport of chaotic 
 chances ; and the study of its laws is the science of art. Rules are merely 
 the expressions of the probable sequences of cause and effect already 
 recognised. They are useful because in many cases they foretell the 
 effect with accuracy. But if their prophecy is false, they must be 
 corrected by a further study of the new effects, under the assumption 
 that the effect is not the result of the one cause stated in the rule, but 
 is the resultant of a number of causes, some of which act in opposition 
 to one another and so produce from time to time apparently contrary 
 effects. 
 
 The prohibition of consecutive fifths appeared comparatively early 
 in the history of the art, much earlier for example than the formulation 
 of the connexions of inversions. In the music of the ancient Greeks, 
 series of fifths or of fourths seem to have been freely allowed in instru- 
 mental, but not in vocal music. This was known as the 'antiphonic' 
 
 w. r. M. 6
 
 82 CONSECUTIVE FIFTHS [OH. 
 
 style. In vocal music only octaves were run in series; no other con- 
 sonance was 'magadised' (16, 21, 154 ff.). On the common instruments 
 of Greek music, the lyre and the cither, sequences of fifths or fourths 
 were only possible in so far as the melos lay within the lower tetrachord 
 of the octave; for in their music the only distinctive mejody lay below 
 the accompaniment. 'After the highest available fourth had been reached, 
 however, the accompaniment remained stationary in the highest tone 
 of the instrument, while the melody wandered at will even into unison 
 with it. When the melody again descended out of this region, it drew 
 the accompaniment with it in fifths or fourths, only the final interval 
 being always the octave (16, 232 ff.). According to Aristotle music 
 involving different intervals ('symphonic' style) was less pleasant than 
 the antiphonic. 
 
 Greek music thus seems to have been essentially monomelodic. 
 Vocal melody was evidently absolutely single (cf. 16, 157). And although 
 in the instrumental style a further approach was made to polyphony, 
 especially when different intervals became obligatory in the upper 
 tones, yet it was clear to the Greek ear that the melos still lay unobscured 
 below the accompaniment. The latter did not itself form a voice (16, 234). 
 
 This state of the art forms a most interesting parallel to the earliest 
 forms of Western music known as organum. In its strict form this 
 consisted simply of series of fifths or of fourths, or of these primary 
 voices doubled at the octave, the upper an octave below and the lower 
 an octave above. The very difficulty that probably prevented the 
 Greeks from magadising in fifths or fourths, namely the occurrence 
 of a tritone instead of a fifth or a fourth once in the complete scale, 
 may have been responsible for the development of a 'free' organum, 
 in which the ' vox principalis ' moved from unison with the ' vox organalis ' 
 up to the fourth while the latter remained stationary, and the like 
 (81, 51 ff.). The variant thus attained was then preferred for its own 
 sake and developed to greater freedom. And after a time the only 
 other possible relation of voices that of contrary motion seems to 
 have appeared quite suddenly (81, 71 ff.) : 
 
 The earliest known expositions of the new organum are contained in the Musica 
 of Johannes Cotto, written about the year 1100.... The organum, we find, is now 
 constructed entirely of consonances, and the arrangement of these is decided chiefly 
 by the various kinds of progression adopted by the voices.... Although the similar 
 [parallel] motion of the voices is by no means forbidden, a contrary progression 
 is on the whole preferred (81, 77). (Hie facillimus ejus usus est, si motuum varietas 
 diligenter consideretur: ut ubi in recta modulatione est elevatio, ibi in organica fiat 
 depositio et e converse. 83, vol. 150, 1429.)
 
 xi] CONSECUTIVE FIFTHS 83 
 
 But the series of consecutive consonances of the same kind did not 
 go beyond two or three. "Existing compositions prove that the first 
 actual expansion of the polyphonic principle, the addition of a third 
 part to the original two, dates from this period, and that the fourth 
 part followed soon after" (81, 85; cf. 44, 79). 
 
 There cannot be the slightest doubt that music in three or four 
 parts in which contrary motion prevails is polymelodic or polyphonic. 
 There is not now any such difficulty in following the various voices 
 as Plato 1 and Aristotle complained of in the Greek music of mixed 
 intervals (16, 149 f.). And it is a noteworthy fact, which our further 
 analysis will illuminate in a far-reaching way, that the decline of the 
 antiphonic style and the gradual emergence of the prohibition of succes- 
 sive fifths and octaves, etc., proceeded in close relation to the develop- 
 ment of distinctive polyphony. 
 
 Thus it appears that musical art can proceed only a little way 
 before it comes to a distinct apprehension of the bad effect of consecutive 
 fifths and before it makes their prohibition a primary principle of con- 
 struction. We are not by any means, however, compelled to suppose 
 that early Greek and Western musicians took perverse pleasure in 
 ill-sounding experiments in symphonious singing. An isolated perfect 
 consonance has at all times a beautiful aspect that reveals itself very 
 readily to the mind, although comparison with some other intervals 
 when they have been found and fully appreciated may make it seem 
 thin and poor. But to the natural uncritical ear a high grade consonance 
 is beautiful. If that beauty is made the object of great attention, it 
 is possible that it might maintain itself for some time in forms of usage 
 that at the same time presented latent aspects 'of ugliness. We have 
 only to suppose that the latter had not yet caught the attention. Besides, 
 this trend of attention would be prevented by another feature of con- 
 sonances the unity of voice or tune to which they in their grade 
 approximate. The voices of men and women singing the same melody 
 will fall into octaves because of the ease and unity thus established. 
 The voices of women or of men, if they differ from one another in pitch 
 considerably, would for the same reason tend to fall into the next 
 greatest consonance the fifth, as the octave would not lie near enough 
 to their average difference to attract their voices to itself; or if it did, 
 one or both voices might be subjected to too much strain. Untrained 
 
 1 Plato perhaps was not specially gifted musically (cf. 66, IT). But it does not require 
 any exceptional musical faculty to follow two simultaneous melodies, if they have been 
 properly composed and performed. 
 
 62
 
 84 CONSECUTIVE FIFTHS [CH. 
 
 singers have been heard to sing in fifths. Stumpf recorded this of two 
 maids at work in his domestic kitchen (52, 239). In primitive music 
 also sequences of fifths have been variously established (ibid.). 
 
 An interesting experiment in such music has been recorded by 
 Gevaert (1895, 15, 423) : 
 
 Sequences of fifths, produced without thirds and performed slowly by very 
 true voices, have nothing disagreeable about them. Consecutive fourths produce 
 at first a bizarre effect, but the ear soon accustoms itself to that. I made a personal 
 experiment with this on the 8th of July, 1871, at an archaeological gathering arranged 
 by my friend Aug. Wagener, the eminent hellenist, in the ruins of the Abbey of St 
 Bavon at Ghent. On this occasion I had a choir of men and children perform several 
 diaphonic specimens of the two species [strict and free Organum]; the impression 
 made on the audience, about a hundred persons, was profound. Everyone was 
 unanimous in finding in this threadbare harmony a penetrating atmosphere of very 
 remote antiquity. It is true that the place lent itself admirably to an evocation of 
 this nature. , 
 
 If we can thus show why sequences of fifths are for some time in 
 the earliest stages of the art not only tolerable, but more or less inevitable, 
 we must endeavour to find out on what basis the unpleasant effect 
 rests that soon appears in the further development of the art. This 
 problem has long been the object of inquiry and debate, and it is well 
 that we should consider carefully what grounds of explanation have 
 already been advanced. 
 
 The following are the chief theories of the prohibition : 
 1. Habit and tradition. This theory was advocated by W. Pole 
 (50, 283 e. ; 17, 113 f.). A ready reason for the prohibition of consecutive 
 octaves is found in the fact that counterpoint is a series of different 
 melodies going together. Using sequences of octaves in counterpoint 
 thus means professing to keep melodies different throughout and yet 
 not doing so. But the rule as to fifths has always been a great puzzle, 
 he says : 
 
 It is asserted and generally believed that there is something naturally repugnant 
 to the ear in such successions.... But still it is undeniable that any series of musical 
 sounds will be accompanied naturally by consecutive fifths as well as by consecutive 
 octaves; and with this example in nature before us, it certainly seems difficult to 
 say that such sequences are forbidden by natural laws. 
 
 We are bound to distrust here the appeal to the ear.... It cannot be denied that 
 a succession of perfect fifths in counterpoint sounds very objectionable to musicians. 
 But it must be recollected that from the first moment any musician began to study 
 composition, he was taught to hold consecutive fifths in abhorrence; and it is to be 
 expected that the result of this must be to make him believe that they are naturally
 
 xi] CONSECUTIVE FIFTHS 85 
 
 objectionable. If there is really any physical or physiological cause for the antipathy, 
 it ought to be capable of being shown; if it cannot be shown, we have a right to 
 presume it is merely the effect of education and habit.... We know one thing by 
 experience, namely, that these fifths do not sound offensive to those who happen 
 to be ignorant of the rule against them. There are many persons who have learnt 
 music practically, and have been accustomed to it all their lives, but who have 
 never had a lesson in harmony or composition; and if such people attempt to write 
 music in parts, they will use consecutive fifths without the slightest hesitation, and 
 not see anything objectionable in them; rather a strong argument, it would seem, 
 that the objection arises chiefly from a knowledge of the rule. 
 
 Even so notable a writer as F. A. Gevaert has given support to a 
 kindred view in writing : " Influenced by the school rule that prohibits 
 the succession of several perfect consonances of the same species, the 
 musicologists have not failed to declare the diaphonies of the epoch 
 of Hucbald and Guido as intolerable and monstrous. That is a counter- 
 pointist's prejudice" (15, 423); and : "it is a modern prejudice to believe 
 that sequences of consonant fifths as such jar on the ear" (16, 158). In 
 support of this he tells of the experiment already quoted (p. 84, above), 
 but he does not add any further justification of his view. 
 
 Pole's theory is, of course, very extreme and may be opposed on 
 every count. As C. Stephens pointed out, the harmonic fifth is so 
 prominent in certain cases, e.g. on stopped organ pipes, which give 
 the alternate harmonics, that it may make that timbre unsuitable 
 with music that would tend to direct attention to its presence, e.g. 
 when a fugal subject is being given out in the lower part of the instrument 
 (17, ns). G. A. Macfarren declared that the sequence of fifths is 
 
 repugnant to us at the present time, and not in this room alone, not in this country, 
 but throughout all the civilised world wherever music is studied, and wherever it 
 has resolved itself into a language instead of the barbarous jargon of savages. I 
 cannot suppose that, as long as the organs of hearing have been the same, persons 
 can have experienced pleasure many hundreds of years ago in progressions which 
 are entirely offensive to us who hear them now; that the same acoustical properties, 
 whatever they may be, which make them offensive in the nineteenth century could 
 have been absent in the tenth century; and that progressions which through their 
 as yet undiscovered properties are cacophonous to us can have been acceptable 
 to the persons who heard them (17, in). 
 
 Of course many persons may have "learnt music practically and may 
 have been accustomed to it all their lives" and yet may never have 
 attended to it analytically or aesthetically at all. Just as there are so 
 many who are hardly even aware of the diurnal changes in the colour 
 of objects in spite of their having attended to these objects in many 
 critical practical ways which would seem to a colour artist to make
 
 86 CONSECUTIVE FIFTHS [CH. 
 
 such ignorance impossible. Think of the crudities such persons would 
 produce in a first attempt at water-colouring ! 
 
 But the weakest point of Pole's position is that, while demanding 
 from the defenders of the prohibition an exposition of the physical 
 or physiological cause for the antipathy, he omits himself entirely 
 even to suggest the need for a cause of the convention, as he thinks 
 it, by which in the first instance sequences of fifths came not only to 
 be forbidden but also to be so heartily disliked. Such a convention to 
 dislike requires as strong a cause as any unauthorised hatred, more 
 especially as opposition to the convention does not seem, even in Pole 
 himself, to have led to a change of taste for consecutive fifths. 
 
 2. Excessive sweetness. According to W. H. Cummings (17, 114) 
 John of Dunstable, an Englishman, forbade consecutive fifths : 
 
 not because they are so objectionable, but because they are so sweet, so that the 
 ancients could be really cloyed with the sweetness of the fifth. We know that fully 
 to the end of the thirteenth century most of the harmony we can find consists of 
 fifths and octaves. They found it so sweet that they thought it was time to leave 
 it off. John of Dunstable is really the first who wrote against the use of them. 
 
 Or as G. A. Macfarren expressed it : " John of Dunstable said they 
 were too beautiful, too much beauty could not be permitted, therefore, 
 a succession of these delights was overpowering to the human senses" 
 (17, us). 
 
 Against this theory Sacchi wrote (57, 6) : 
 
 No one ever denied, nor shall I, that successive sweetness can change to 
 displeasure. We can therefore well understand that a continued series of ten or fifteen 
 fifths ought to displease and disgust us; and that would not be improbable. But 
 that one single repetition of the fifth, merely by reason of its great charm, must 
 suddenly displease and offend us, is not intelligible. 
 
 That is quite true. But we have not only to refute the theory, but 
 to account for its formulation as well. And it is not difficult to see the 
 motive of it. The fifth, as was noted above, is the second best consonance 
 or fusion, and as such has a special beauty and sweetness, not necessarily 
 under all circumstances of comparison, but at least under some. Of 
 these circumstances probably only the latter group will control the 
 judgment of the primitive ear and of any mind that is for the time 
 being more or less uncritical or forgetful of the specially interesting 
 intervals and chords that music has developed. In any case there is 
 no doubt that the primitive ear is fixed upon the beauty of the great 
 consonances, and naturally endeavours to make its art out of these
 
 xi] CONSECUTIVE FIFTHS 87 
 
 elements. No doubt too for a time its attention to them prevails over 
 any other features their sequences may create. But, of course, these 
 features, being so pronounced, soon force themselves upon the attention. 
 What is more natural, then, than that the only known feature of these 
 intervals their great consonance should be taken as the ground of 
 explanation, and that the theory should be : in sequences of fifths 
 there is too much of the fifth's consonance? It is not that the ears of 
 these early folks were undeveloped or different from ours, and that 
 their minds were crude and lame; but their attention had been set 
 into a certain direction by the course of the art till their time ; and their 
 minds naturally followed its suggestions. Looking backward is not 
 nearly so difficult as groping forwards and accommodating soul and 
 mind to new developments quickly. We must judge leniently when 
 we think how long all our theories have tried to nourish their energies 
 on the very poor diet of the harmonics. 
 
 3. Want of variety. Zarlino wrote in 1571 (82, Part iii, chap. 29, 
 p. 216) : 
 
 The most ancient composers forbad the placing after one another of two perfect 
 consonances of the same genus and species bounded in their extremes by one and 
 the same proportion, while the modulations moved by one or more steps; as the 
 placing of two or more unisons, or two or more octaves, or two or more fifths, and 
 such like ; . . . for they well knew that harmony cannot spring but from things mutually 
 diverse, discordant, and contrary, and not from such as in every way agree. 
 
 The composer, he says, must imitate the beauty of nature which makes 
 no two things of any species exactly alike. 
 
 The explanation given by Helmholtz includes this one beside others : 
 
 The accompaniment of a lower part by a voice singing an octave higher, merely 
 strengthens part of the compound tone of the lower voice, and hence where variety 
 in the progression of parts is important, does not essentially differ from a unison. 
 Now in this respect the nearest to an octave are the twelfth, and its lower octave 
 the fifth. Hence, then, consecutive twelfths and consecutive fifths partake of the 
 same imperfection as consecutive octaves (20, 359). 
 
 Only, the case is worse because the accompaniment cannot be carried 
 out consistently without changing the key. (The a of the key of c 
 is familiarly a little flatter than the just a of the key of d: thus two 
 just fifths c-g and d-a would mean a departure from the scale of c in 
 the a.) "Hence an accompaniment in fifths above, when it occurs 
 isolated in the midst of a polyphonic piece, is not only open to the 
 charge of monotony, but cannot consistently be carried out" (20, 360).
 
 88 CONSECUTIVE FIFTHS [CH. 
 
 Helmholtz then proceeds to explain that : 
 
 when the fifths are introduced as merely mechanical constituents of the compound 
 tone, they are fully justified. So in mixture stops of the organ.... It would be quite 
 different if we collected independent parts, from each of which we should have to 
 expect an independent melodic progression in the tones of the scale. Such independent 
 parts cannot possibly move with the precision of a machine; they would soon betray 
 their independence by slight mistakes, and we should be led to subject them to the 
 laws of the scale, which, as we have seen, render a consistent accompaniment in 
 fifths impossible. 
 
 For the same reasons the second inversion of the major common chord 
 "expresses a single compound tone much more decidedly than 'the 
 first inversion,' which is often allowed to be continued through long 
 passages, when of course the nature of the thirds and fourths varies" 
 (20, 360). The second inversion may be represented as the third, 
 fourth, and fifth partials of a compound tone, the first inversion as 
 'only' the fifth, sixth, and eighth. But, as F. E. Gladstone pointed out 
 (17, 102), this argument will not apply to minor chords. The minor 
 chords have always been a thorn in the flesh of the harmonic derivations 
 of music. 
 
 The theory, then, argues that close approximation to the constitu- 
 tion of a tonal blend of fundamentals and partials makes sequences 
 of fifths and fourths admissible. These sequences are forbidden only 
 between distinct parts, because we expect independence and variety 
 from them, not monotony (cf. Gladstone, 17, 105). The theory suggests, 
 but does not state explicitly, that the prohibition of consecutives is 
 the stricter the nearer the interval in question lies to the fundamental 
 component of a blend. Thus the fourth is prohibited, "but with less 
 strictness" than the fifth. "Even thirds" have been forbidden as an 
 accompaniment (cf . ibid.). The theory is apparently consistent logically. 
 
 What is hard to understand is how the relation to partials creates 
 such unpleasantness in this case while in single consonances it makes 
 for harmony and pleasantness. Of course every theory must appear 
 in dealing with this problem to pull from the storehouse of explanation 
 contrary results for what seem very similar objects. That is a mere 
 restatement of the fact that isolated consonances are pleasant, while 
 sequences are often ugly. What every theory, however, must avoid 
 doing is using the same unaltered ground as an explanation of contrary 
 results. And that Helmholtz seems to do. If "hearing again a part of 
 what we heard before" is a ground of consonance, it is unlikely that 
 this alone would produce the ugliness of sequences of fifths. Monotony 
 is an idea hardly adequate to the effect to be explained. As Sacchi
 
 xi] CONSECUTIVE FIFTHS 89 
 
 might have said : we could understand that ten or fifteen fifths in 
 sequence would have been boredom, but we should hardly take offence 
 at two. Helmholtz seems to have felt this himself somewhat; for he 
 inclines in part to Pole's view, saying : 
 
 The prohibition of consecutive fifths was perhaps historically a reaction against 
 the first imperfect attempts at polyphonic music, which were confined to an accom- 
 paniment in fourths or fifths, and then, like all reactions it was carried too far, 
 in a barren mechanical period, till absolute purity from consecutive fifths became 
 one of the principal characteristics of good musical composition. Modern harmonists 
 agree in allowing that other beauties in the progression of parts are not to be rejected 
 because they introduce consecutive fifths, although it is advisable to avoid them 
 when there is no need to make such a sacrifice (20, seo). 
 
 Here he has not reached the point of view of some who claim that 
 in these exceptional cases it is not a matter of admitting the ugliness 
 for tne sake of the beauty, but of outweighing the ugliness so that it 
 no longer appears or even of creating positive beauty (cf. 35, 84). 
 
 The argument from want of variety is weak in so far as it has to 
 meet its own objection for e\ery interval; including sixths which by 
 no manner of means can be claimed as forbidden in sequence (cf. 
 C. Stephens, 17, H5, and G. A. Macfarren, 17, 119, who refer to both 
 thirds and sixths). Sacchi added that 
 
 it is true that on the false principle that two fifths displease for lack of variety, 
 some [e.g. Zarlino 1 ] have drawn the false conclusion that similarly the repetition 
 of thirds and of sixths ought to displease when they are of the same species, and 
 have therefore forbidden it; but their prohibition was not accepted by composers, 
 who, disciplined by experience, carefully avoid repeating fifths, but are not in the 
 least concerned about the repetition of thirds or sixths of the same species. Vain 
 is therefore likewise the reason that is drawn from the desire for variety, which 
 if true would be equally so in all the consonances; it would hold rather more in 
 the imperfect than in the perfect; because after all it ought to be more tolerable 
 to the ear to linger on the sweeter consonance than on the less sweet (57, ^ t.). 
 
 The validity of the argument, therefore, vanishes entirely in so 
 far as it is mere variety. Variety is certainly desirable, but it would 
 be as pedantic to prescribe it at every instant as to forbid consecutive 
 fifths when they sound well. 
 
 If by variety we mean specially the monotony of compound tones, 
 
 1 Non si debbe anco porre due 6 piu imperfette consonanzo insieme ascendent! 6 dis- 
 cendcnti 1' una dopo 1' altra senz' alcun mezo; come sono due Terze maggiori, due minori, 
 due Seste maggiori anco e due minori. Conciosiache non solo si fi contra quello c'ho 
 dctto delle Perfctte; ma il loro procedere ei fa udire alquanto aspro; per non haver nella 
 lor modulatione da parte alcuna 1' intervallo del Semitonio maggiore, nel quale consisto 
 tutto '1 buono nclla Musica, e senza lui ogni Modulatione cd ogni Harmonia 6 dura, 
 aspra, c quasi inconsonante [82, 217).
 
 90 CONSECUTIVE FIFTHS [CH. 
 
 surely the condensation of the partial components into the range of an 
 octave would make a sufficient variation from the compound tone. 
 And the inevitable departure from the justness of the fifths when the 
 music remains consistently in one key would help to assuage the 
 monotony. Consecutive fifths would then partly cease to be fifths, 
 and should in so far be tolerable. Besides, if two consecutive fifths of 
 different pitch involve too little variety and are therefore forbidden, 
 is there not still less variety in the repetition of one and the same fifth ? 
 And yet that sequence is unobjectionable. 
 
 Thus it appears that the argument is insufficient to explain the 
 effect produced by two consecutive fifths. Some much more decided 
 difference must be the source of the ugliness in question. 
 
 We shall see as we proceed that, if the word 'independence,' which 
 has often been used in this connexion instead of the word 'variety/ 
 were properly emphasised and defined, much could be said in favour 
 of the theory. The reader may therefore in the end feel that the variety 
 theory has much in its favour. No doubt those who advocated it felt 
 this underlying justification, but in their expressions they refer only 
 to independence in the sense of variety or difference of voices. They 
 did not mean by independence the independence, as distinct from the 
 mere variety, of the voices. 
 
 4. Ambiguity of key or tonality. One of the theories discussed by 
 Sacchi falls more or less under this head, although it is not quite the 
 same as the form most familiar at the present time. The prohibition 
 of consecutive fifths in this case was held to be due to the great difference 
 between the scales that arise from the bases of the successive chords. 
 Thus the scale of d differs from that of c in two notes (the two sharps). 
 
 But, said Sacchi, why not imagine in the scale of d instead of a major, 
 a minor, third (as in the melodic minor scale). Then there would be 
 a difference of only one note, i.e. the least possible difference, between 
 the scales. And, after all, the difference alleged is not really heard, 
 it is only imagined, or at least conceived. The argument surely attributes 
 too great force to the imagination. Besides, the four tones heard 
 (c-g and d-a) actually belong to one and the same scale and have 
 optimal consonance with one another. How then could they be turned 
 into an offence by mere imagination or rather by the mere possibility 
 of imagining two other notes? "This will certainly in no wise happen : 
 for things imagined, and that can be, never prevail over such as are 
 or are felt" (57, 8ff.).
 
 xi] CONSECUTIVE FIFTHS 91 
 
 Although the theory in question would hardly find a champion 
 nowadays, the argument against the force of imagination is noteworthy. 
 Imagination or the inclination of interpretation is often very important 
 in music, but it is well to be reminded that such a fluctuating and 
 divertible force is not likely to be the cause of a very constant and 
 highly undivertible phenomenon, such as the unpleasantness of con- 
 secutive fifths. 
 
 The theory of the prohibition advanced by Sacchi himself is the am- 
 biguity of tonality created by the sequence of fifths, e.g. c-g and d-a. 
 
 I have no sufficient reason to refer the second note (d) to the first (c) or conversely 
 the first to the second (57, 76). I might be told that I ought to refer the second 
 ut to the first, because the sound of the first has already taken possession of my ear. 
 But with equal reason I might be told that the first ut ought to refer to the second, 
 because all other things being equal, the present sensation, being more lively, is 
 to be given precedence of the past. The suspension is therefore unrelieved; and I 
 am lost in ambiguity, indetermination and suspense between two different notes, 
 each of which, without any difference, can be considered as primary base (ibid. f.). 
 
 Those who follow this type of theory do not usually admit that all 
 other things are so perfectly equal as Sacchi said. And if they are, is 
 there not an obvious means of overcoming the ambiguity by indicating 
 the key with all possible precision before the consecutive fifths are 
 introduced? Then the ground of objection to them alleged by Sacchi 
 would be removed. If imagination is an insufficient force to explain 
 the offensiveness in question, we must surely conclude that any such 
 ambiguity is only very slightly more potent. An easy way out of any 
 such difficulty is in constant evidence in every exactly measured sequence 
 of objectively equal intervals of time. The attention may elect to 
 hear this sequence as one or other of various rhythms : w ', w ', w ' . . . or 
 w j v ^j v w or x ww > ' w ^ N v ~ and so on. But in spite of the great 
 number of possibilities, each of which can be realised at inclination, the 
 ear never remains tortured by ambiguity. It adopts at once a definite 
 rhythm, possibly the easiest under the circumstances. When it has 
 had enough of that one, it 'fluctuates' into another, it may be. If the 
 answer be that such an involuntary solution is impossible in dealing 
 with keys because they are different from mere rhythms, then we must 
 reply that if the apprehension of tonality is at all difficult and not 
 so inevitable as is rhythm, then the mind should receive the sequent 
 chords without any 'thought' of tonality at all, as an unmusical mind 
 certainly would. 
 
 And Sacchi himself excludes the last possible reply at this point 
 by his words :
 
 92 CONSECUTIVE FIFTHS [CH. 
 
 In fact the two successive fifths not only offend the ear of the erudite in music, 
 but of those even who have no practice in it, so long as they happen to have been 
 born with a good and subtle ear, and pay attention to what they hear. For where 
 one does not attend, which of the dissonances will not pass unobserved and without 
 offending? (57, ie). 
 
 The central attitude to chords, whereby the lowest component is 
 the most sonorous, most before the attention, etc., may well be inevitable 
 even to those unpractised in the ways of music; but no one could well 
 suggest that the apprehension of tonality is such an inevitable and 
 'natural 5 attitude, requiring no practice and experience. There is some 
 evidence that the Greeks related the pitches of all their notes to one 
 particular note the mese, at least for the purposes of tuning, if not 
 with some feeling for 'tonality.' In the latter case the functions of their 
 tonic must at least have been very different from ours. Before a certain 
 period of modern music the sense of tonality was much weaker than 
 it is now. In various cases it may have hardly been present at all. 
 
 Finally, ambiguity of tonality is by no means uncommon in more 
 modern music at least. Transposition from one key to another is 
 frequently affected by means of one or more chords that are common 
 to both keys. In a familiar piece of music, then, one may not only ' hear ' 
 the key that is to be left but one may be able to anticipate that to come, 
 so that the transitional chords may be in a real sense ambiguous. And 
 yet no such horrid effect is thereby produced as is characteristic of 
 consecutive fifths. Considerable dispute is often possible as to which 
 key a short passage of a musical piece really displays; but no specially 
 inartistic effect appertains to such passages. Mere ambiguity of key 
 without any other difference is therefore useless as an explanation of 
 the prohibition of two sequent fifths. 
 
 Perhaps no explanation is more frequently offered for the disagreeable effect 
 of consecutive fifths than that suggested by Cherubini: viz. that two parts moving 
 progressively by fifths are moving in two different scales 1 . The reason is obviously 
 insufficient; but it has more force than some critics are willing to admit, when, for 
 instance, three triads in succession are based upon the notes C, D, E, the first having 
 
 1 For a very primitive form of this theory in the writer of the Commentary called 
 Scholia Enchiriadis, see 81, wt.: "We learn that it is the impropriety of this combination 
 of two different modes or species of the scale, throughout the whole of a composition, 
 which in his view gives rise to the necessity for a free treatment [of the organum]." 
 (Quare in Diatessaron symphonia vox organalis sic absolute convenire cum voce princi- 
 pali non potest, sicut in symphoniis aliis ? Quoniam per quartanas regiones non iidem 
 tropi reperiuntur, diversorumque troporum modi per totum ire simul ire nequeunt. 83, 
 vol. 132, 1003; cf. p. 972.)
 
 xi] CONSECUTIVE FIFTHS 93 
 
 a major third, and the other two having minor thirds, but each with a perfect fifth, 
 it seems clear that the parts do not all progress in one scale throughout. The upper 
 parts cannot really be in the scale of C, because, as we know, neither a true fifth 
 nor a minor third can, strictly speaking, be based upon the second degree of an 
 accurately tuned major scale. But it may be said that the instrument upon which 
 these chords have been played, is tuned. upon the system known as 'equal tempera- 
 ment.' No doubt it is. Nevertheless I contend that, as we tolerate its sharp major 
 thirds and flat fifths, knowing and feeling them to be substitutes for the true thirds 
 and fifths of the genuine scale, so we are accustomed to accept other divisions of 
 the scale, not for what they actually are, but for what they represent.... To me, 
 therefore, it does not seem unreasonable to argue that even with the pianoforte 
 we recognise the equivocal nature of such a progression as that contained in my 
 first example [triads in C, D, E}. But even after this has been granted, the argument 
 that consecutive fifths cause two parts to move in different scales cannot be carried 
 much further. The triads on the third, fourth, fifth, and sixth degrees are all 
 perfectly in tune in the scale of just intonation. Some further reason, therefore, 
 must be sought for the unquestionably ugly result produced when these chords 
 are taken in regular rotation, either ascending or descending (17, ioof,; cf. G. A. 
 Macfarren (17, iwi.; 35, 10). 
 
 The argument thus properly refuted by F. E. Gladstone contains 
 at the best something very like a dilemma which renders it obscure 
 and confusing. When we hear fifths true to the scale of c major on 
 c, d and e, our musical habit may make us do either of two things : 
 either we are governed by our habit of the major scale on c, and then 
 the intervals are heard as played and the second is not a true, but only 
 an approximate, fifth and is heard as such; or we are drawn away 
 rather by the habit of the perfect fifth, when the intervals will not be 
 heard as they are played, but only as they suggest in perfect form and 
 the sequence will lie in no one scale. If the former alternative is valid, 
 the prohibition of consecutive fifths must hold for all fifths, whether 
 perfect or approximate, if the ugly effect persists in spite of the recognised 
 approximation (and that it does persist could hardly be denied). The 
 alternative would only prohibit such fifths as lead to a distortion of 
 the intended scale. A third alternative might claim that we hear all 
 the tones and intervals as played, but that we take the approximate 
 fifths as representing true ones and are disturbed by the distortion 
 of scale thereby implied. This would surely be a needless procedure 
 in view of the fact that even if we take approximate thirds as representing 
 just thirds (minor), we do not thereby feel any distortion of scale. 
 
 Of these three alternatives the first seems not only easier and more 
 natural but also most in accordance with the whole system of musical 
 synthesis and apprehension. It is the only one which makes the system
 
 94 CONSECUTIVE FIFTHS [CH. 
 
 of equal temperament musically tolerable for permanent use or at 
 least as for any individual the only known system. For the other 
 alternatives presuppose that the pianoforte is only tolerable in virtue 
 of the just scales it suggests, which must have been otherwise in- 
 eradicably planted in the minds of all those who use that instrument. 
 That is probably true of those whose experience has made them most 
 familiar with instruments that play just intervals. But for the great 
 majority the intervals their minds apprehend when the piano is played 
 are those actually played, the scale known is the pianoforte scale, and 
 the rules and prohibitions of musical structure are valid for all these 
 approximations. 
 
 It is important to notice that these approximations are primarily 
 matters of consonance and its grades. An equal fifth partakes as a 
 consonance of the grade of fusion found optimally in the just fifth. 
 The same holds still more for the dissonances. And as intervals the 
 latter can be learnt in their approximate form; the just form of the 
 dissonant intervals has no such special nature as a fusion as would 
 enable it to draw the ear towards itself. It is a familiar fact that the 
 high grade consonances have this power. It is difficult even to rtrike 
 the dissonances that lie near these consonances because the voice tends 
 to slip into the easier and more familiar consonance. 
 
 A special objection to the key theory lies in the fact that consecutive 
 fifths are only objectionable when they lie between the same voices 
 of the music. That is a primary condition of the phenomenon, but it 
 is by no means a condition of sameness of key. In music of any definite 
 tonality all the voices of any moment are held, and are intended, to 
 be in the same key, whatever that may be. There is no recognised 
 complication of tonality in which a group of keys are considered to be 
 maintained concurrently, one in each separate voice, or pair of voices. 
 Hence the key theory loses its ground entirely. 
 
 It is true that in recent compositions parts or groups of parts have 
 sometimes been made to move concurrently in different keys (cf. 23, 
 138 f.). But that is quite a different matter. Each group of parts is 
 still subject to the fundamental rules of harmony, although as between 
 the separate groups the claims of these are largely ignored. 
 
 5. Want of relationship. The theory advanced by Gladstone is 
 "that consecutive fifths are generally more or less offensive in proportion 
 to the want of relationship, or otherwise, existing between the chords 
 which produce them." He cited Kollmann as probably the first writer
 
 xi] CONSECUTIVE FIFTHS 95 
 
 who propounded this idea 1 . Unfortunately his case was spoilt at the 
 very outset by his admission that there is as little relationship between 
 the inversions of chords on successive notes of the scale as between 
 their original positions: and of the inversions a succession of six- three 
 chords is quite admissible. The objection attaching to the six-fours 
 does not alter this fact in the least, nor its destruction of the theory 
 of relationship. We may, therefore, expect the frequency with which 
 consecutive fifths are admitted between tonic and dominant chords 
 and vice versa, or between tonic and subdominant chords and vice 
 versa, to have some other cause. We cannot interpret this frequency 
 as due solely to the high degree of relationship between the chords in 
 the sense of relationship implied when we say that the triads on C and 
 D are totally unrelated. Gladstone admitted besides, that he had "not 
 met with any specimens of consecutive fifths in which the roots of the 
 chords rise a third (except where a sudden change of key occun)" 
 (17, 104), and evidently had found only two cases of a fall of a third 
 tonic to submediant. 
 
 Gladstone proceeded to note that the objection might be raised 
 that this argument ought also to apply to fourths, thirds, and sixths; 
 and recalled in reply that the movement of fourths is placed under 
 various restrictions by the laws of counterpoint, and that even two 
 major thirds in succession are still forbidden in the strictest style of 
 two-part writing (17, 105). But however interesting this extension of 
 the basis of argument may be, it is perfectly obvious that the unexplained 
 relaxation of the prohibition in these cases only makes the theory of 
 relationship the more impossible. 
 
 6. The nature of the interval itself. The study of the connexion 
 between consecutive fifths (and octaves) and the harmonic relationship 
 of chord? has been renewed by Shinn (58, 205 ff . ; 59). But Shinn 
 does not attempt to explain the prohibition of successive fifths and 
 octaves by the lack of harmonic relationship between the chords in 
 which they stand, but by the intrinsic character of the intervals of the 
 
 1 It was also the basis of Pearsall's (1795-1856) explanation who wrote that 
 "consecutive great thirds and perfect fifths are evidences that some harmony has been 
 sprung over which ought to have been introduced by its characteristic note, as forming 
 the natural link of relationship between these intervals." When they are not evidences 
 of such a spring, "they carry with them an awkwardness of progression which ought 
 to be avoided." They display "a want of freedom and a clumsiness, unacceptable to any 
 musical ear" (49, as). But not even strong disapproval forms a logical complement to 
 an incomplete theory.
 
 96 CONSECUTIVE FIFTHS [CH. 
 
 octave and the fifth themselves. For the octave Shinn adopts the 
 explanation that may be termed traditional. The bad effect of a ' hidden ' 
 octave and a fortiori of consecutive octaves is "the weakness which 
 is produced by the correspondence in sound of the two outside parts 
 (approached in this manner) " (58, 268). This statement revives the 
 explanation by want of variety discussed above. The bad effect of a 
 hidden fifth, and a fortiori of consecutive fifths is due "to the bareness 
 of the interval" (58, 268, 280). 
 
 This expression may be taken either as a variant upon the 
 "weakness" of the octave or as a new kind of reason that has not been 
 advanced by any other theorist, as far as I am aware. It makes perhaps 
 some approach towards the notion of consonance as approximate unity. 
 But obviously it is now an aspect of the interval (or fusion) of the 
 fifth that displeases, whereas in the octave it is the correspondence in 
 sound of the two tones that make up the interval the so-called identity 
 or similarity of octave-tones. Such a difference of causes could hardly 
 be acceptable. 
 
 It is not easy to give a just systematic place to Shinn's exposition. 
 In some ways he suggests the first theory of this chapter. He says, 
 for example (58, 263) : 
 
 Combinations and progressions which were formerly regarded as painfully crude, 
 harsh and ugly, have, by familiarity, lost these characteristics, and become both 
 piquant and pleasant; while others, which had hitherto produced pleasure, now 
 seem commonplace in comparison with the poignancy of less familiar but more 
 forcible ones. In connexion with this matter, the important point to be recognised 
 is, that no change has taken place in the progressions themselves, but it is the ear 
 of the listener which has changed, owing to the influence of a change in his musical 
 environment (58, aes). 
 
 Elsewhere he speaks of the "so-called objectionable effect of con- 
 secutive fifths" (p. 284). 
 
 On the other hand he points out that this bad effect (whether so- 
 called or not) is "almost invariably neutralised by the harmonic 
 relationship which exists between the chords forming such fifths" 
 (ibid.}. The bareness of the fifth, we are to understand, is somehow 
 annulled or enriched. But Shinn neither explains why the fifth is a 
 bare interval I suppose it just sounds so nor does he show how chord 
 relationship removes this bare character from the interval. In fact he 
 is not always quite faithful to the explanation by harmonic relationship 
 and abandons it in part in favour of "the effect of musical strength 
 which is characteristic of (such) progressions" of the two voices con-
 
 xi] CONSECUTIVE FIFTHS 97 
 
 cerned by a fourth and a fifth respecti\ely (58, 277). It is not entirely 
 a matter of "harmonic relationship (or root progression), but partly 
 also of the movement of the outside parts by almost equal intervals." 
 The latter acts even without any special harmonic relation. 
 
 Nevertheless Shinn's discussion is, as we shall see later, probably 
 the most 'philosophical' one that has so far been given. 
 
 7. Want of balance. A suggestion made by G. A. Macfarren is 
 worthy of mention, although it has not been worked out into a definite 
 theory as far as I am aware. "When a passage of harmony in any 
 number of parts has two notes made so very much more prominent 
 than the rest, as is the case in the duplication of those two at the expense 
 of the others, the other portion of the harmony is enfeebled, and the 
 balance is destroyed" (17, us). This is exemplified in the case of 
 successive octaves, when two notes mutually reinforce each other and 
 so become particularly prominent over against the rest of the score. 
 The same does not hold for the doubling of a voice, even although the 
 relation is now merely two to three instead of two to two; for here the 
 doubled part is meant to be specially prominent. 
 
 In this restricted form the theory of balance is not very significant. 
 For the balance in question is chiefly a balance of mere strength. An 
 overbalance of strength can easily be produced in music and often 
 occurs not only by mere accident and through the imperfect technique 
 of performers, but through want of finish on the part of the composer. 
 On none of these occasions could it well be said to be so strikingly 
 unpleasant as to justify the view that consecutive octaves (and fifths) 
 are forbidden so strictly because of the disproportion of strength they 
 produce. 
 
 w. F. M.
 
 CHAPTER XII 
 
 THE SYSTEM OF FACTS REGARDING CONSECUTIVES 
 
 THE preceding chapter contained a review of the explanations that 
 have been attempted for the prohibition of consecutive fifths. We have 
 noticed how the various theories try to set the prohibition into relation 
 to facts of a similar kind so as to obtain some indication of systematic 
 coherence in the explanation. None of the systems of facts thus suggested 
 is very satisfactory or convincing; and none of the theories can possibly 
 be held to be successful. It is extremely doubtful whether anyone who 
 has thus far reflected on the problem of consecutive fifths has felt that 
 more than interesting suggestions towards an explanation have been 
 reached. 
 
 "Often and often have I thought," said G. A. Macfarren, "it would 
 require the entire knowledge of a physicist to be able to probe this 
 subject to its foundation" (17, 119). But the day when physical science 
 may be expected to solve such a problem is now definitely past. Even 
 Helmholtz, whose basis of explanation might well seem to many to be 
 physical, was perfectly well aware that the ground of explanation of 
 all musical phenomena must lie within the phenomenal stuff of sound 
 itself; it dare not be merely physical (20, 231 f., 368). No doubt the 
 prominence, or at least the great propinquity, of the physical throughout 
 his exposition prevented many of his followers from giving sufficient 
 heed to the psychical or, if you like, to the phenomenal aspect of the 
 problems of music. Besides the difficulty and apparent obscurity of 
 the psychical itself made them only too eager to seize upon any plausible 
 excuse for evading the study of its elementary aspects. Such an excuse 
 was not only given, but even emphasised by Helmholtz. 
 
 The system of scales, modes, and harmonic tissues does not rest solely upon 
 inalterable natural laws, but is also, at least partly, the result of esthetical principles, 
 which have already changed, and will still further change, with the progressive 
 development of humanity (20, 235). 
 
 This proposition, he said, was "not even now sufficiently present 
 to the minds of our theorists and historians." But ever since then, at 
 least, it has been decidedly obstructive in its effect upon their minds. 
 It was a most unfortunate dictum. For the opposition implied in it 
 between natural laws and aesthetical principles strongly suggested that
 
 CH. xii] REGARDING CONSECUTIVES 99 
 
 the latter are merely arbitrary conventions, as is more or less, for example, 
 the fashion of clothes in any year. After quoting this dictum Prout 
 wrote : 
 
 While, therefore, the author [himself] follows Day and Ouseley in taking the 
 harmonic series as the basis of his calculations, he claims the right to make his own 
 selection, on aesthetic grounds, from these harmonics, and to use only such of them 
 as appear needful to explain the practice of the great masters (52, 1st ed., 1889 iv). 
 
 And many others besides Prout could be quoted to the same effect. 
 
 But an aesthetical principle is not the sort of thing that men for 
 centuries in vain seek to explain. So hidden a cause is rather an aesthetical 
 law, which is just as much law as is any physical uniformity. And it 
 can no more be laid aside in this arbitrary way than an ethical standard 
 can be suppressed whenever you think it will not approve of what 
 you choose to do. 
 
 A noticeable feature of these attempted explanations of consecutive 
 fifths is their fragmentariness and isolation. Most theorists give only 
 a short statement of what seems to them to be an easy and obvious 
 reason for the prohibition of octaves, namely, the disturbance they 
 produce in the balance or in the melodic distinctiveness of the parts. 
 And some theorists refer to the minor restrictions placed upon sequences 
 of fourths (and even of thirds) in confirmation of the different theory 
 they offer for fifths. It may seem to many minds quite satisfactory 
 to have one solution for the octaves and a second for the fifths and 
 other intervals. Difficulties that are allowed to slumber, of course 
 make no attack. But there still remain the few intervals that are not 
 prohibited in succession at all. No one who has a keen sense for the 
 systematic logic of a theory can long remain satisfied with such work. 
 And so the problem of consecutive fifths remains to-day without any 
 recognised solution. 
 
 Once the psychical ground of the phenomena of music has been 
 recognised, it may seem to be an inevitable consequence that no satis- 
 factory or convincing explanation of such phenomena can be given. 
 There are many who think the appeal to the subjective judgment 
 necessarily unconvincing. In his introduction to his account of Rameau's 
 doctrines D'Alembert wrote : 
 
 Here must not be sought that striking evidence that is peculiar to works of 
 geometry and that is so seldom met with in those in which physics mingles. There 
 will always enter into the theory of musical phenomena a sort of metaphysics that 
 these phenomena implicitly suggest and that brings thither its own natural obscurity; 
 in this matter we must not look for what is called demonstration; it is much to have 
 
 72
 
 100 THE SYSTEM OF FACTS [CH. 
 
 reduced the principal facts to a system well linked and well pursued, to have deduced 
 them from a single experiment, and to have established on this so simple basis the 
 best known rules of musical art. But, on the other hand, if it is unjust to exact 
 here that intimate and unassailable persuasion that is produced only by the most 
 vivid light, at the same time we doubt if it is possible to throw a greater light upon 
 these matters (9, xiii f.). 
 
 Since D'Alembert's time even the demonstrations of physics, that 
 have become so numerous as to be a sort of standard for all sciences, 
 have been subjected to such searching examination as to make some 
 minds incline to see in them only a complete description of events. 
 No doubt there is much more involved in them than this. But that 
 more is itself the source, not of a superiority of physics to the science 
 of the foundations of music, but on the contrary of a kind of philosophical 
 inferiority. For physics implies the positing of many types of real 
 entities the substantial basis of the phenomena that are so perfectly 
 described. At least all but a very few thinkers allow this feature to 
 enter freely into their physical constructions. Only a few extremists 
 shall we say? such as Ernst Mach and Bertrand Kussell, have attempted 
 to claim that physics as a science may be construed without any such 
 postulations, but merely by description or by classification (of pheno- 
 mena) as a fundamental process. 
 
 The science of music, however, has as it were its whole perspective 
 in converse form. Its facts are obviously phenomenal; it not only begins 
 with the completest description, but, in the opinion of many, it neces- 
 sarily ends there too. For music, they hold, is entirely phenomenal. 
 Of course, when the basis of explanation is carried back into the physical 
 realm, as after sufficient description and explanation of the phenomena 
 themselves it properly may, our knowledge of the basis of music 
 then goes beyond the bounds of the phenomenal realm. That, however, 
 is not the point at issue. Those who say the basis of music is entirely 
 phenomenal mean that its whole task is necessarily mere description. 
 Description and classification, and the study of sequence and of 
 dependence amongst phenomena cannot, they think, lead to any know- 
 ledge of phenomena that could show them to be not wholly phenomenal. 
 
 Only very few venture to claim that the science of musical phenomena 
 may gain knowledge of these phenomena that shows them to be more 
 than phenomena, to be at least partly real, entities independent of our 
 minds that do not necessarily reveal themselves completely and finally 
 to us at the first glance or after any amount of inspection. In so far 
 then, as the science of music passes so rarely or never over the frontiers 
 of the phenomenal, its descriptions can proceed with fewer questionable
 
 xn] REGARDING CONSECUTIVES 101 
 
 or uncertain assumptions, and may be looked upon as more completely 
 defensible in a logical sense than can even the work of so highly successful 
 a science as physics. 
 
 However that may be, there is no doubt at all nowadays that the 
 science of phenomena can attain to as complete description of its objects 
 as any science, and can thereby compel conviction as completely. No 
 doubt the differences to be described are often very subtle; but they 
 may often be clear and easy to distinguish. In any case we must not 
 nourish false expectations. Phenomena cannot, for example, be magni- 
 fied with microscopes. We are at the very outset already at the limits 
 of possible magnification. But apart from this sort of thing, the methods 
 of a science of phenomena are as reliable and to those whose minds 
 are open to conviction as convincing as are those of any science of 
 nature. 
 
 There are indeed very many at the present time whose minds for 
 various reasons have closed completely against this idea that a sufficient 
 science could be made of mere phenomena such as is the heard stuff 
 of music. They would not deny that an art could be raised upon this 
 basis. But art they may feel as subjective and variable with the caprice 
 and inclination or even with the ' personality ' of each man. Nevertheless 
 we must insist upon it that the more an art is studied, the more it is 
 felt to be a realm of order and coherence. No doubt in a science of art 
 we are dealing with the finer, more intimate, issues of events, and not 
 so much with the great lines of Nature's efforts. But in arts there are 
 also broad beams of construction. And the natural sciences have all 
 already come into contact with the subtlest and finest issues of their 
 objects, so that this difference between the sciences of Nature and of 
 Art has no longer any effective validity with reference to their dignity 
 as systems of knowledge. 
 
 The first task that presents itself in every problem of the foundations 
 of music is to describe the phenomena as completely as possible. 
 Preliminary to this is the effort to get all the phenomena together. That 
 must be done by starting from the phenomenon that first raised the 
 problem, for example, our problem of the prohibition of consecutive 
 fifths, by searching for all phenomena in any degree similar to this 
 striking one, and by endeavouring to find a systematic arrangement 
 and description that will incorporate them all. 
 
 That is the logical status of the method of solution. It may in 
 some cases be the method of discoverv as well. As a matter of fact the
 
 102 THE SYSTEM OF FACTS [CH. 
 
 systematic arrangement now to be expounded was first suggested by an 
 attempt to reduce the rules for part- writing given in E. Prout's Harmony 
 (52) to comprehensive and facile form by making a table of the objects 
 shown by the rules to be of chief importance octaves, fifths, fourths, 
 sevenths, seconds, and ninths along with the recurrent factors that 
 seemed to modify their admission or prohibition, e.g. the different 
 voices, the kind of motion, etc. This effort suggested its own extension 
 and completion to what seems highly probable as at least a close 
 approximation to the system of facts of which consecutive fifths form 
 a part and from which a satisfactory explanation of their prohibition 
 may flow. This system of facts seems, moreover, to fall into place as 
 an extension of the system of facts regarding the foundations of music 
 already expounded. Not only so but it seems also to renew their ground 
 in an independent manner, which may lend great weight both to the 
 facts as already described and to the systematic description and explana- 
 tion they have received. 
 
 Thus the series of intervals just mentioned may be completed by 
 the addition of thirds and sixths. Then we have octaves, sevenths, 
 sixths, fifth, tritone as diminished fifth or as augmented fourth, fourth, 
 thirds, and seconds. We shall omit consideration of the so-called prime 
 or of unison for the present (cf. below, p. 112). These are al] the intervals 
 smaller than the octave. If we arrange them in their order of fusion, 
 we get : octave, fifth, fourth, thirds and sixths, tritone, sevenths and 
 seconds. For the study of consecutives in general this order is of much 
 greater importance than is the former. 
 
 Of the circumstances that modify the prohibition of consecutives 
 the series of voices is one of the most important. Let us begin with 
 the consideration of four-part harmony. The bass, as we have already 
 seen, is the naturally predominant voice. The next in order is the 
 soprano, partly because it is the other outside voice, and partly because 
 it usually bears the most important, if not the only (coherent or thema- 
 tised) melody in harmonic music. It is the only voice that is often claimed 
 to be more noticeable in a single stationary chord than is the bass 
 (cf. p. 51 ff., above). There is no obvious distinction in importance be- 
 tween the two other voices. This grading of the voices is confirmed and 
 greatly strengthened by the way in which the stringency of the rules 
 of part- writing is relaxed on occasion in relation to the different voices. 
 
 But each interval must necessarily involve two voices at once, so 
 that the series bass, soprano, tenor or alto, has to be squared with 
 itself, so to speak. The grading that ensues is : (1) bass-soprano, (2) bass-
 
 XII] 
 
 REGARDING CONSECUTIVES 
 
 103 
 
 alto or bass-tenor, (3) soprano-alto or soprano- tenor, (4) alto- tenor; 
 or, as we shall often find it convenient to write them : B-S, B-A and 
 B-T, S-A and S-T, A-T. 
 
 When this series is correlated with the series of intervals just stated, 
 Table I results. No definite preconceived idea determined the form of 
 it. The idea was merely to arrange the chief objects referred to in the 
 rules of part- writing as given by E. Prout (52, 25 ff.) and their chief 
 relations so that any system implicit in them might become patent. 
 
 TABLE I 
 Consecutives (preliminary system) 
 
 Showing relations between (1) the stringency of prohibition (Forb. - = almost 
 strictly forbidden; forb. = forbidden with exceptions; forb. - = with more exceptions; 
 + = allowed) and (2) the grade of fusion or consonance, and (3) the prominence 
 of the voice-parts. Based upon the formulations of E. Prout (52, sad.). 
 
 
 Octaves 
 
 Tritone 
 to 
 fifth 
 
 Fifths 
 
 Fourths. 
 Tritone to 
 fourth 
 
 Thirds 
 or 
 sixths 
 
 Sevenths 
 
 Seconds 
 or 
 ninths 
 
 B-S 
 
 Forb. - 
 
 Forb. 
 
 forb. 
 
 forb. 
 
 + 
 
 forb. 
 
 Forb. 
 
 B-A 
 
 Forb. - 
 
 Forb. 
 
 forb. - 
 
 forb. - 
 
 + 
 
 forb. 
 
 Forb. 
 
 B-T 
 
 Forb. - 
 
 Forb. 
 
 forb. - 
 
 forb. - 
 
 + 
 
 forb. 
 
 Forb. 
 
 S-A 
 
 Forb. - 
 
 forb. 
 
 forb. - 
 
 + 
 
 + 
 
 forb. 
 
 Forb. 
 
 S-T 
 
 Forb. - 
 
 forb. 
 
 forb. - 
 
 + 
 
 + 
 
 forb. 
 
 Forb. 
 
 A-T 
 
 Forb. - 
 
 forb. 
 
 forb. - 
 
 + 
 
 + 
 
 forb. 
 
 Forb. 
 
 Tritone is the diminished fifth or augmented fourth, as the case may be; it is 
 not reckoned as a fourth or a fifth when it follows a fourth or a fifth respectively. 
 The Table has been arranged so as to bring out a grading in the stringency of 
 prohibition from below upwards and from side to side. 
 
 The rules upon which this Table is based are the following (52, 25 fl.) : 
 
 (1) No two parts in harmony may move [in unison, or] in octaves with one 
 another. There is one exception to the prohibition of consecutive octaves. They 
 are allowed by contrary motion between the primary chords [tonic, dominant, 
 sub-dominant] of the key, provided that one part leaps a fourth and the other a 
 fifth. 
 
 (2) Consecutive perfect fifths by similar motion are not allowed between any 
 two parts. They are, however, much less objectionable when taken by contrary 
 motion, especially if one of the parts be a middle part and the progression be between 
 primary chords [T.D.Sd.]. This rule is much more frequently broken by great
 
 104 THE SYSTEM OF FACTS [CH. 
 
 composers than the rule prohibiting consecutive octaves. Consecutive fifths between 
 the tonic and dominant chords are not infrequently met with. 
 
 If one of the two fifths is diminished, the rule does not apply, provided the 
 perfect fifth comes first.... But a diminished fifth followed by a perfect fifth is 
 forbidden between the bass and any upper part but allowed between two upper 
 or middle parts, provided the lower or occasionally the upper part moves a semitone. 
 
 (3) Consecutive fourths between the bass and an upper part are forbidden, 
 except when the second of the two is a part of a fundamental discord [whose intervals 
 are a major third, a perfect fifth, and a minor seventh from the generator 1 (39, 94)] 
 or a passing note, i.e. a note not belonging to the harmony. Between any of the 
 upper parts consecutive fourths are not prohibited. They are sometimes found 
 between the bass and a middle part; but even these are not advisable. 
 
 (4) Consecutive seconds, sevenths and ninths are forbidden between any two 
 parts, unless one of the notes be a passing note.... There is one important exception 
 to this rule to be found in the works of the old masters. Corelli, Handel, and others 
 sometimes followed a dominant seventh by another seventh on the bass note next 
 below. 
 
 There is some difference of opinion amongst authorities as to the 
 special digressions from the prohibitions that are admissible. But if 
 we take account chiefly of the existence and degree of freedom of 
 exceptions we may look upon Prout's rules as relatively valid. 
 
 Thus Parry (45) says that "there are so many consecutive sevenths 
 to be found in the works of the greatest masters, and that, when they 
 are harsh, they are so obviously so, that the rule prohibiting them 
 seems both doubtful and unnecessary." Here the point of view is 
 mainly practical. The ugliness of the sequence is really admitted, 
 although in a restricted form; and so it appears in our Table. Text- 
 books of harmony evidently find it unnecessary to state that consecutive 
 thirds and sixths are unobjectionable. But the fact is of the greatest 
 importance as a datum for theoretical work for all that. It is just these 
 obvious facts that no one mentions that are often the keystone of a 
 successful theory of such phenomena. 
 
 There is evidently a system inherent in the Table. For we see that 
 the two outside columns contain nothing but Forb. or Forb. , i.e. 
 strict or almost strict prohibitions for all voices. In the second column 
 the prohibition is relaxed a little in the lower three pairs of voices 
 (forb.). In the third column that relaxation, and perhaps even a little 
 more of it (forb. ), holds for all the pairs of voices except B-S. In 
 the fourth column further progress is made in the same direction; the 
 sequence is admitted for the lower three pairs of voices. In the case 
 
 1 The 'generator' is the lowest note of the so-called root-position of the chord.
 
 xn] REGARDING CONSECUTIVES 105 
 
 of the thirds and sixths there is perfect freedom. But the prohibition 
 comes into force again in a restricted degree for the sevenths, and is 
 finally complete for the seconds and ninths. The grading of the voice- 
 pairs that makes this system possible is identical with the grading 
 deduced above from the experimental and general facts of analysis 
 of chords. 
 
 The most significant and suggestive feature of the Table, however, 
 is the sequence of the intervals seen in the top horizontal column. 
 This sequence is very nearly the same as that already given for the 
 grades of consonance of intervals that lie within the octave. And the 
 result suggests strongly, so far at least as this Table shows the situa- 
 tion. that the degree of stringency of the prohibition of consecutive 
 intervals of the same species depends (1) upon the grade of consonance 
 or dissonance, and (2) upon the prominence of the two voices that 
 constitute the intervals in question. The Table thus takes proper 
 notice of the fact that the consecutive intervals must both lie between 
 the same two voices. 
 
 A special problem is created in the Table by the tritone. As a 
 so-called diminished fifth it is not forbidden when it follows, but only 
 when it precedes, a perfect fifth. The same holds in so far as it is reckoned 
 as an augmented fourth in connexion with a perfect fourth. The prohibi- 
 tion of the tritone when the fifth follows it, seems to be stricter even 
 than the prohibition of two fifths. This extreme difference between the 
 two successions shows that we are here not dealing with consecutive 
 intervals of the same species. When the augmented fourth or diminished 
 fifth follows, it is not a fifth or a fourth at all, but another kind of 
 interval at least as far as the system of facts represented in the Table 
 is concerned. It is for that reason it has been classified in the Table 
 as a tritone. Thus the problem of the tritone reduces itself to the single 
 case in which it precedes either a fifth or a fourth. This is obviously 
 not an instance of consecutive intervals of the same kind. In respect 
 of the fifth alone it belongs to the case of 'hidden fifths.' Otherwise 
 the succession of tritone and fifth belongs to the class of problems 
 that includes all such questions as : under what circumstances may 
 any two intervals of different species follow one another in the same 
 voices? 
 
 We may, therefore, remove the tritone from its present position 
 in the preliminary Table of consecutives and replace it properly, having 
 regard solely to the degree in which consecutive tritones are avoided
 
 106 THE SYSTEM OF FACTS [CH. 
 
 or forbidden in part-writing. Text-books of harmony contain no state- 
 ment regarding consecutive tritones, either as diminished fifths or as 
 augmented fourths. But in dealing with modulation a practical oppor- 
 tunity occurs of presenting information on this subject. Thus P. 
 Tchaikovsky says (76, 70 f.): 
 
 There are also sequences in which every chord constitutes a modulation. They 
 are those in which dominant seventh chords or other chords resolving into the 
 tonic succeed one another, always falling a fifth or rising a fourth, as in a sequence 
 within the limits of one key. In such a sequence each chord resolves into a chord 
 which itself demands resolution and forms at the same time the resolution of its 
 precursor. 
 
 He then gives progressions containing five to eight tritones in 
 succession, diminished fifths alternating with augmented fourths. Of 
 course there is some difference between these two intervals in their 
 musical significance. But in respect of their fusion and even in respect 
 of their specific nature as intervals (proportions of volumes) there is 
 practically none. In one of Tchaikovsky's examples there is even a 
 series of eight simultaneous pairs of tritones (chords of the diminished 
 seventh), one tritone lying between the two outer voices, the other 
 between the two inner voices. Prout (52. 1st ed., 162) gives an example 
 from Bach's Chromatic Fantasia in which successive tritones abound. 
 
 We may, therefore, look upon successive tritones as being- more 
 freely admissible than consecutive sevenths, major or minor. The 
 diminished seventh, which may be run in succession (52,242) is practically, 
 and from the point of view of fusion, quite the same thing as the major 
 sixth, which is not restricted at all. No doubt the musical affinities of 
 intervals that are apprehended as diminished sevenths will call for a 
 different treatment of them from that of intervals apprehended as major 
 sixths. But it is clear that in dealing with consecutives we are not 
 concerned with such special apprehension of musical setting and relation- 
 ship, but with a more fundamental matter that appertains to the 
 intervals in question almost in any setting, so long as they lie between 
 the same voices. 
 
 We may, therefore, place the tritone in an amended Table of. 
 Consecutives between the thirds, sixths and the sevenths. 
 
 This brings the final Table into much greater conformity with the 
 experimentally established grading of fusion of the different intervals. 
 In fact so far as the differentiation of our Table shows, in which the 
 different thirds or sixths or sevenths, etc., are not distinguished, the
 
 XIl] 
 
 REGARDING CONSECUTIVES 
 
 107 
 
 conformity is complete. The most frequent grading of fusion that 
 experimental research has as yet shown is (77, 104) : 0, 5, 4, III, 3, VI, 
 6, T, II, 7, 2, VII (cf. above, p. 16). 
 
 TABLE II 
 Consecutives (Final System) 
 
 Showing relations between (1) the stringency of prohibition (Forb., Forb. -, 
 forb., forb. -, + or allowed), and (2) the grade of fusion of any interval, and (3) the 
 prominence of the voice-parts in which the interval appears. 
 
 
 O's 
 
 5's 
 
 4's 
 
 3's & 6's 
 
 T's 
 
 7's 
 
 2's & 9's 
 
 B-S 
 
 Forb. - 
 
 forb. 
 
 forb. 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 B-A 
 
 Forb. - 
 
 forb. - 
 
 forb. - 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 B-T 
 
 Forb. - 
 
 forb. - 
 
 forb. - 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 S-A 
 
 Forb. - 
 
 forb. - 
 
 + 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 S-T 
 
 Forb. - 
 
 forb. - 
 
 + 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 A-T 
 
 Forb. - 
 
 forb. - 
 
 + 
 
 + 
 
 forb. - 
 
 forb. 
 
 Forb. 
 
 The Table shows that the grades of fusion from greatest consonance to greatest 
 dissonance in relation with the relative prominence of the pair of voices on which 
 the interval in question rests give rise to a system of preferences or prohibitions 
 of a very well graded kind (cf. p. 103, above). 
 
 And the following conclusion may be drawn. If due consideration 
 is given to the prominence of the pair of voices that bear the interval 
 in question, it appears that the immediate repetition of an interval 
 in the same voices is the more offensive the greater the consonance or 
 dissonance of that interval. The point of minimal unpleasantness or 
 of maximal pleasantness (as the case may be) in the series from greatest 
 consonance to greatest dissonance lies amongst the thirds and sixths. 
 These intervals may, therefore, be held to be fusionally neutral. 
 
 This inference differs a little from the prevalent attitude towards 
 the thirds and sixths. They are nowadays ranked among the consonances. 
 They are, of course, certainly not dissonances. But, on the other hand, 
 they are perhaps not really consonances either. 
 
 It is a familiar fact that the ancient Greeks did not include them 
 amongst their consonances, which were octave, fifth, and fourth, alone, 
 and stated in this order by Aristoxenus and by Ptolemy (cf. 66, 38, 58). 
 This fact has often been interpreted as indicating that the Greeks 
 considered the thirds and sixths to be dissonances, as we now understand 
 this term. But that may not be taken for granted. The system indicated
 
 108 FACTS REGARDING CONSECUTIVES [CH. xn 
 
 in Table II suggests strongly that the thirds and sixths may not have 
 been included amongst the consonances by the Greeks because they 
 are not appreciably 'positive degrees of consonance 1 . That we find them 
 highly pleasant and characteristic is not at ail inconsistent with the 
 correctness of this estimate. 
 
 Of course the mere fact that we rank the thirds and sixths after 
 the fourth in the grading of fusions that lead from greatest consonance 
 to greatest dissonance, implies nothing at all as to whether these intervals 
 are consonances or dissonances. And the experimental evidence 
 regarding the very slight percentage difference between the grades (of 
 approximation to the impression of a single tone) lower than the fourth, 
 and th.e similar evidence regarding the variations in the serial arrange- 
 ment of the grades of fusion show that at least there is no clear division 
 between these lower grades. So if a minor seventh is a dissonance, a 
 minor third can hardly be a strong consonance; nor can even a major 
 third. There must be a point at which dissonance passes into con- 
 sonance. Logically that point may be a vanishing point, of course. 
 But even then the lower consonances, if we suppose the thirds and 
 sixths to be positive consonances, must have a very low degree of 
 consonance to be so slightly different from the lesser dissonances and 
 so often confusible with them in respect of fusion. It, therefore, seems 
 probable that the grades of fusion including the thirds and sixths may 
 properly be considered to be neutral. 
 
 Thirds and sixths, then, are neither distinct dissonances, nor are 
 they distinct consonances. And the Table of Consecutives gives us a 
 very strong reason for accepting this description. For the treatment 
 there shown to be accorded to thirds and sixths is distinctly different 
 from that accorded to the extreme consonances and to the extreme 
 dissonances. 
 
 1 Cf. Gevaert (14, 102): "Let us notice first that the meaning of the terms has been 
 modified in the course of time. We translate symphonia by consonance, diaphonia by 
 dissonance. So did even the Romans in the Augustan age, always attaching to these 
 words another idea than we do. The fundamental difference distinguished by the ancients 
 between the two kinds of intervals is that in symphony the sounds fuse to a perfect unity, 
 whilst in diaphony they maintain their individuality and detach themselves in some way 
 from one another. In this respect our impression does not differ appreciably from that 
 of the Greeks. For us too the thirds and sixths have a clear cut character that is lacking 
 in the fifth and fourth. We notice the same clearness in the second and in the seventh; 
 and from this new point of view we find it possible to let the ranking of these two intervals 
 in the same category as the thirds and sixths pass." Gaudentius was the first to admit 
 the major third (and also the tritone) amongst the consonances. To the former inclusion 
 we now generally agree, but only with special effort to the latter (cf. 14, 99; 66, 7i f.).
 
 CHAPTER XIII 
 
 THE REASON FOR THE PROHIBITION OF CONSECUTIVES 
 
 THE question which next arises is the one from which all previous 
 writers on the subject have started : why are consecutives offensive? 
 Why are all these consecutives offensive, each in its degree? 
 
 The system of facts we have discovered in the preceding chapter 
 does not answer the question directly. It only arranges the objective 
 facts with which any answer to the question must reckon. It indicates 
 that the solution must not only be the same for octaves as for fifths, 
 but it must even be the same for both dissonances and consonances. 
 The only thing common to them all is some degree of fusion. The 
 repetition of a high grade consonance or dissonance introduces a new 
 and special feature that is unpleasant. Thus our task must now be to 
 show a basis for this feature and to form a theory as to its nature which 
 will adequately justify on conceptual grounds the unpleasantness of 
 the effect we hear. 
 
 The problem may be approached in two ways. In the first, experi- 
 mentally, we might present a series of observers with a systematically 
 varied complex of consecutive pairs, and ask for direct observation and 
 description of the feature of each that is unpleasant or more or less 
 preferable. This course would certainly not be successful in such a 
 simple form, although it would be of great value for the grading of the 
 intervals on the basis of their preferability. Consecutive intervals of 
 the same species have been considered and compared and reflected 
 upon for centuries already without even any indication of agreement 
 having been reached as to what it is in the sequence that is directly 
 unpleasant. Even those who have written treatises on the subject 
 have hardly done more than guesswork upon the problem, except in 
 so far as they attempted to infer a basis of unpleasantness from the 
 system of facts gathered round the central object of inquiry. 
 
 The situation in this particular aspect of it is similar to that of the 
 theory of consonance and dissonance. In spite of the fact that the 
 ancient Greeks and the older writers of the modern era had defined 
 consonance as the mixture or blending of two tones into one, that 
 direct description did not receive in the more modern explanations by 
 the relations of the harmonics the central importance due to it. It was
 
 110 THE REASON FOR [CH. 
 
 only restored to its proper position by the critical studies of Stumpf. 
 And even Stumpf could not go beyond this amount of direct description, 
 already attained by the Greeks, to say more definitely and decisively 
 how the fusing tones interpenetrated one another so as to approximate 
 to the effect of a single tone. If it was almost impossible in this case 
 to proceed beyond the terms of direct description to an adequate theory 
 of the basis of the phenomenon in the sensory stuff of the tones them- 
 selves, how can we expect by direct observation to win a theory of the 
 bad effect of consecutives, seeing that even the direct description of 
 that effect has not yet been obtained? 
 
 Evidently the work of observation and description must be facilitated 
 by the discovery of definite alternative questions, to be answered by 
 the comparison of minor differences. In other words, we must learn 
 how to instruct the observer so as to make description easier for him 
 by directing his attention precisely towards possible special features 
 of consecutives. We must expect a properly instructed and careful 
 course of systematic observation to confirm any inferences as to the 
 basis of the prohibitions that may otherwise be gathered. 
 
 For that is the other way of approaching the problem. By enlarging 
 the system of facts in which consecutive fifths stand, we have already 
 obtained a much better formula for their prohibition than we could 
 have obtained from a study of them alone. The fifths are forbidden 
 because of something that emerges from pairs of highly positive or 
 negative fusions. Perhaps if we enlarge in turn the system of facts 
 of which consecutives form a part, we may attain some still more 
 specific formula. Armed with this, we could return to the work of 
 direct description with some hope of obtaining a definite answer to a 
 definite question. A probable further system of facts suggests itself 
 in the well-known counterpart to consecutives the prohibition of 
 single intervals, commonly termed hidden octaves and fifths. 
 
 But in the facts already before us there is an aspect that calls for 
 some notice, although it may at first glance seem trivial and obvious : 
 that the intervals to be prohibited must lie between the same voices. 
 Of course that is not equivalent to saying that they must lie somewhere; 
 they might lie between different voices, and when they do so, they give 
 rise to no feature that is objectionable. Evidently when we listen to 
 music in several parts, our attention even in music that is pre- 
 dominantly harmonic runs along the voices, as it were, noticing the 
 series of relations that emerge between the successive tones of each
 
 xni] THE PROHIBITION OF CONSECUTIVES 111 
 
 pair of voices. It does not connect into systems one relation between 
 one pair of voices, a second relation between another pair of voices, 
 a third relation between a third pair, and so on 1 . The systems are rather 
 those that actually present themselves serially. The relations in question, 
 however, are fusional, or, as Hullah said, perpendicular relations. Only, 
 chords are not apprehended even in music that is predominantly 
 harmonic as unanalysed wholes; the apprehension is not fusional or 
 perpendicular throughout the chord as an undivided unity. Analysis 
 breaks this whole into parts of two voices at least; these are the units of 
 fusional or perpendicular apprehension. 
 
 In polyphonic music this much is also undoubtedly true. Only here 
 there enters another factor that justifies the contrary term 'horizontal 
 apprehension,' namely the distinctively melodic or thematic treatment 
 of each voice. The figures, forms, and phrases of melody are maintained 
 in each voice over and above the restrictions that are placed upon 
 their progressions by the fusional aspects of pairs of voices. It is not, 
 then, so much the case that harmonic music has introduced a feature 
 not yet present in polyphonic music; but rather in the latter there is 
 present in highly cultivated form a feature which prevents the harmonic 
 relations implicit within it from coming into prominence. No doubt, 
 too, this suppression prevented these relations from being specially 
 cultivated. But, whether cultivated or not, they are essentially present 
 in both types of music. 
 
 Thus we obtain some closer specification of the relations between 
 synthesis and analysis in music generally. In listening to music in several 
 parts we do not apprehend the fusions of chords in so far as they 
 approximate to the balance and symmetry of a single tone as a. whole 
 mass. Our attention is always, up to a certain degree, analytic. We 
 notice always the relations between pairs of voices. And to do so we 
 must be able to maintain the proportions of the volumes as defined by 
 each pair of voices in the forefront of our attention. For that purpose 
 analysis is necessary. 
 
 Now we have a perspective from which to judge the generalisation 
 attained from Table II. A succesiion of high grade consonances or 
 dissonances is very unpleasant; it is offensive according to the degree 
 of consonance or dissonance (or of fusion positive to the neutral grades 
 
 1 So the crossing of parts will obviate consecutive fifths that appear when the parts are 
 in pianoforte score (cf. 52, 104 1. for example). But such voice-leading will, of course, require 
 the support of a difference in blend between the voices, as in choral or chamber music.
 
 112 THE REASON FOR [CH. 
 
 or negative to them) and to the prominence of the pair of voices con- 
 cerned. In short, prominence of high or low grade fusion disturbs; 
 neutral grades of fusion do not disturb. Disturb what? Only one 
 answer suggests itself : they disturb (the analysis or the set of attention 
 required to maintain) the usual flow of presentation of relations between 
 the pairs of voices. The horizontal view, so far as it is generally attained 
 in music, is disturbed by the undue prominence of the perpendicular 
 relation between the voices. Either the voices interpenetrate too much 
 in successive pairs so as to cut off the connexion between the two tones 
 of either voice (thus the connexions c-d and g-a are broken in con- 
 secutive fifths on c and d}\ or the voices disrupt from -one another too 
 markedly and thus also break the connexion unduly within each voice. 
 Neutral grades of fusion alone do not in succession break this even 
 flow of analytic concentration necessary for the appreciation of the 
 greater works of music 1 . 
 
 We cannot go into the further aspects of this formulation at once. 
 We must await the systematic arrangement of the facts included in 
 these further aspects, keeping the formulation before us as a hypothesis 
 to be tested and enriched. But it is at least evident now why con- 
 secutives are forbidden only in connexion with a movement of the voices. 
 The repetition of any interval without any change of its pitch would 
 in no way affect the apprehension of the sequent tones of each voice. 
 For as nothing has changed, the attention has for the moment an 
 easier task than usual. On the other hand a succession of unisons tends 
 to betray the analytic attention into losing hold of the individuality 
 or duple nature of the voices that thus temporarily coincide. One 
 unison, however, is not disturbing so long as the voices are felt 
 melodically to converge and to coalesce ; and if the next chord is suitable, 
 they will be felt to separate and to diverge again (cf. below, p. 130). 
 
 1 Descartes explained the prohibition of consecutive octaves and fifths thus: "Ratio 
 enim quare id magis expresse prohibeatur in his consonantiis quam in aliis, est quia hae 
 sunt perfectissimae; ideoque, dum una ex illis audita est, tune plane auditui satisfactum 
 est. Et nisi illico alia consonantia ejus attentio renovetur, in eo tantum occupatur, ut 
 advertat parum varietatem et quodammodo frigidam cantilenae symphoniam. Quod 
 idem in tertiis aliisque non accidit: immo, dum illae iterantur, sustentatur attentio, 
 augeturque desiderium, quo perfectiorem consonantiam expectamus" (11, 132). The com- 
 parison with the case of thirds is interesting. But it serves only to bring out the older 
 point of view which concentrated on the perfect consonances, and not the modern point 
 of view in which the thirds play the more essential part. Descartes' theory is of the 
 'variety' type. That, however, is true only as an approximation towards what variety 
 makes possible and what is attainable in some cases (e.g. with thirds) even without variety, 
 namely continuity of melody.
 
 xm] THE PROHIBITION OF CONSECUTIVES 113 
 
 It must also be now clear that concurrence of voices in fifths or 
 fourths is only tolerable in a primitive stage of music. There the homo- 
 phonic interest is almost the only one present in the music. Polyphonic 
 relations have either not yet been attained at all or only on rare occasions, 
 so that even when they do occur, the ready dispositions of the hearer's 
 mind will not easily yield to any unpleasantness they may bring when 
 consecutives appear. Both singers and hearers intend and know the 
 concurrent voices to be the same melody. No doubt they have to will, 
 or to attend to, the melodic continuity more energetically when they 
 use consecutive fifths and fourths than when they use octaves or a 
 bare melody. But they may be quite willing to do so for the sake of 
 the variety thus attained, until further variation and closer attention 
 show them that the bad effects thus ignored have no compensating 
 power to please by heightening contrast; or that, if they have this 
 power, the systems of variation made possible by it are so small and 
 weak as compared with the systems of variation admitted by changes 
 of consonances, and especially by the use of the lower degrees of con- 
 sonance, that they are not worth while, or are not profitable lines of 
 development, and so are best barred out altogether. Hence their 
 gradual disappearance from music as it progressed towards the form 
 and style of distinct polyphony, and their vigorous prohibition until it 
 had developed enough to allow of their re-introduction amongst many 
 minor systems of variation in a way that does tend to enrich the 
 structural potentialities of music. 
 
 On the other hand the octave does not make melodic continuity 
 at all difficult to maintain so long as the presence of the intention to 
 such continuity has been made evident or so long as the intention 
 towards melodic diversity has not been declared. The reason for this 
 freedom is not so much the fact that the octave is the first harmonic 
 of a fundamental, whereas the fifth is the second. For that should 
 only establish a gradation of difficulty, as it does perhaps in the primitive 
 mind, not a difference between pleasure and offensiveness, as it does 
 in our music. The reason is rather that in the systems of intervals of 
 our music or in our tonality the octave is the absolute basis of reference 
 of all intervals, and is so because of the fact that the increase of an 
 octave means the decrease of volume by half, and because this difference 
 does not alter or distort any pattern of volumic proportions (cf. above 
 p. 72 ff.). A tone and its octave are therefore very easily apprehended 
 as one thing, and that unit of pattern may be followed with great ease 
 throughout all sorts of changes of its volume as a whole. The doubling 
 
 w. F. M.
 
 1U REASON FOR PROHIBITION OF CONSECUTIVES [CH. xm 
 
 of a melody in octaves, then, is admissible in our music because it is 
 quite easy to follow melodically and it is quite consistent with the 
 volumic structure of our systems of intervals. 
 
 It has sometimes been said that the reason for the prohibition of 
 consecutive octaves was that the effect of a four-part harmony was 
 thereby lost. It is now evident from our system of facts and from the 
 place of the octave in it that the reason cannot be of this merely negative 
 order. The consecutive octaves must present a big positive something 
 that is offensive. We shall form a clearer idea of what this is as we 
 proceed.
 
 CHAPTER XIV 
 
 EXCEPTIONS TO THE PROHIBITIONS OF CONSECUTIVES 
 
 IN arranging the system of facts regarding consecutives we had to be 
 content with an approximation to agreement in the statement of the 
 rules. It was enough to bring out the general trend of the differences 
 included within the system without striving to define it exactly in its 
 absolute form. Fortunately there is not very much difference of opinion 
 in the statements of these rules given in the chief text-books of harmony. 
 In arranging the rules which state exceptions to the prohibitions of 
 consecutives we shall again have to rely upon some estimate of the 
 main trend. The general and growing agreement amongst theorists 
 will facilitate our work for the present. 
 
 A full and sufficient account of these rules and their exceptions 
 would best be based upon a very extended statistical treatment of the 
 musical material. No doubt some of the theorists who have worked 
 out rules of prohibition and of admission have collected large numbers 
 of instances and have based their generalisations upon them. But for 
 the fullest understanding much would be gained from an analytic 
 study and a statistical manipulation of such a collection, if it were 
 published in an extended form, so that the reader might follow the 
 relative quantitative importance of the various factors that are found 
 in groups of exceptions. What is required for an elementary knowledge 
 of the principles of construction has doubtless already been attained. 
 But even in some elementary matters these formulations have become 
 detailed enough to show considerable divergence of opinion. This 
 divergence is possibly not so much a sign of any difference in aesthetic 
 reaction between persons or of any aesthetic differences in their nature, 
 as rather a result of the consideration by each of them of different 
 special groups of exceptions, or of only some of the factors operative 
 in typical cases in abstraction from other accompanying ones that 
 contribute to the final aesthetic effect. In any case there can be little 
 doubt that an analytic study of a large number of exceptions on a 
 statistical basis would be of great service both to the science and to the 
 art of music. This sort of effort may be commended to those who have 
 any favourable opportunity for making such large collections. 
 
 82
 
 116 EXCEPTIONS TO THE [CH. 
 
 The exceptions admitted are few. For octaves (and fifths) Macfarren 
 (35, 82f.) stated that the use of the sequence, "however rare, by com- 
 posers of the present century, proves that this most stringently 
 proscribed progression may produce an effect of measureless beauty, 
 when it lies between the chord of the tonic and either 'dominant or 
 subdominant,' provided only that, in the case of octaves, the parts 
 that have the two in succession proceed by contrary motion " (cf . A. Day, 
 10, 58). And Prout, as quoted above, agreed to this exception in the 
 second edition of his work on harmony. Macfarren illustrates the point' 
 from Beethoven's Pastoral Symphony (between the bass and the alto) 
 and from Sonata, Op. 53 (major common chord with doubled root in 
 each hand on dominant and then on tonic). Similarly Tchaikovsky 
 notes that in strict part- writing "(fifths and) octaves are permitted 
 in the inner voices if contrary motion be employed" (76, us). Parry (45) 
 points out that consecutives are most objectionable in vocal and 
 chamber music; in pianoforte and orchestral music they are often lost. 
 
 Shinn discusses the question at some length (58, 276f.), and claims 
 that the sequence of octaves in contrary motion or of octave and 
 unison often produces "an exceptionally strong musical effect." He 
 generalises beyond the tonic-to-dominant or subdominant relation 
 stated by Macfarren towards "other pairs of triads standing in a similar 
 relationship with regard to their progression such as the triads upon 
 the mediant and submediant," etc., "but these are not often employed." 
 The musical strength of these progressions is not due "entirely to the 
 fact of their harmonic relationship (or root progression), but partly 
 also to the movement of the outside parts by almost equal intervals 
 that is, one by a fourth and the other by a fifth." The effect when one 
 part moves a third and the other a sixth, especially when the bass 
 moves the sixth, is generally less strong. Consecutives can rarely be 
 employed in a satisfactory manner when one part moves a second and 
 the other a seventh. It is not clear whether Shinn means the harmonic 
 relationship of the chords or the mere movements of the voices to be 
 the more important element in the effect; probably the latter. 
 
 Octaves in similar motion are admissible according to their purpose 
 and position. As examples Shinn gives one (from Beethoven) for "the 
 emphasising of a full cadence by the outside parts moving from dominant 
 to tonic," another for the formation of a special melodic figure, and two 
 between the final and initial chords of two sections. "In this position," 
 he says, "their employment is by no means rare." 
 
 Shinn does not propose to sanction consecutive octaves when they
 
 xiv] PROHIBITIONS OF CONSECUTIVES 117 
 
 occur in connexion with discords. Here we find further verification of 
 the greater power of the octave towards bad effect. 
 
 As regards fifths, Prout's rule 1 may be taken as a generally accepted 
 nucleus. His statement that the rule for fifths is much more frequently 
 broken by great composers than the rule for octaves is well borne out 
 by the relative frequency of examples to be found in text-books of 
 harmony. From the various writers I have consulted (17, 23, 35, 38; 
 45, (48), 58, 61) I have collected some fifteen examples of octaves and 
 over sixty examples of fifths. This relation may seem somewhat strange 
 in view of the fact that it is customary to speak of consecutive octaves 
 rather lightly and as being objectionable merely because of the 
 temporary loss of distinction between the two voices 2 . But it is obvious 
 that this theory was merely a deduction from the notion of the musical 
 equivalence of octaves; it did not properly reflect the nature of the 
 musical phenomenon itself. And some writers even proceed to explain 
 the bad effect of fifths by the loss of independence of the voices they 
 appear in. 
 
 As regards the influence of the progression referred to by Prout, 
 Gladstone wrote that "of the various exceptions which the great 
 composers have made to their rule of avoiding fifths, none are more 
 common than those in which the progression is either from the tonic 
 to the dominant, from the tonic to the subdominant, or the reverse 
 of either" (17, 103f.). This is the first stage of his argument in favour 
 of explaining the effect of fifths by the relative position of the chords 
 in which they occur. It is again to be regretted that his statements 
 were not accompanied by some evidence showing relative frequencies. 
 For the next degree of relationship (a third between the roots) he could 
 only cite two cases with an ascent of a sixth between the roots and none 
 with a third. 
 
 In connexion with fifths Shinn (58, 280 ff.) seems to rely entirely upon 
 harmonic relationship, making no allusion to the movements of the 
 voices. He indicates a decreasing frequency of occurrence and a loss of 
 
 1 Of contrary motion A. Day wrote: "Fifths by contrary motion should not be used 
 (although by most writers allowed), as the reason given why fifths by similar motion 
 should not be used [they give the idea of two different keys] is equally applicable to fifths 
 by contrary motion " ( 10, 10). So much the worse for the reason given, one might rather say ! 
 
 1 Pearsall, for instance, dismisses consecutive octaves and unisons in five lines (of 
 his 27 page quarto pamphlet), saying they ought to be avoided because of ' awkwardness,' 
 and "because they produce no effect except that of rendering insipid and almost nullifying 
 any harmony of which they may be component parts" (49, M).
 
 118 EXCEPTIONS TO THE [CH. 
 
 effect in the series of relationships of a fifth or fourth, a third or sixth, 
 and a second or seventh. In the last case the effect is rarely satisfactory 
 except for special purposes in the root position of the chords; it is 
 better in inversions. Shinn gives many examples of sequences of fifths 
 in connexion with discords 1 (essential and unessential) and suspensions. 
 The explanation he offers of these is that the dissonant note imparts 
 to the chords "such a new, distinctive, and relatively speaking, forcible 
 character, that the unpleasantness due to the consecutive fourths 
 [fifths], if it is not obliterated, is neutralised by the introduction of 
 this new element and the sound of the progressions becomes entirely 
 satisfactory. Not only is this explanation perfectly adequate, but 
 it is, we believe, the only one which it is possible to supply that is 
 based upon the musical effect of such progressions" (58, 286f.). 
 
 This last exception claimed by Gladstone and Shinn is confirmed by 
 the large number of instances of it that are to be found in writers on 
 harmony. I have collected and compared more than sixty cases of 
 consecutive fifths, mostly from the greatest and most accepted com- 
 posers 2 . Of these 11 show a progression from a discord (commonly 
 a minor seventh) to a concord, 16 from a discord to a discord, and 15 
 from a concord to a discord. In reckoning these numbers I have counted 
 as one only one type of progression in a given work. Thus a seventh 
 to a seventh would reckon as one, no matter how often it were repeated ; 
 but a seventh to a common chord ending the passage would count as 
 a new case. Only five of the 42 cases are by contrary motion. Sixteen 
 lie between the bass and the soprano (B-S), 10 are B-T, 8 are S-A, 
 7 are A-T, and only one is S-T. In resolving these sevenths naturally 
 pass sometimes to dominant and to tonic; but this is a feature of the 
 progression that must be considered accidental ; it is not what legitimates 
 the succession of fifths. 
 
 That can only be the discord in question. And the effect produced 
 is a reasonable one 3 . In this type of exception the two ends of the 
 
 1 Day held that fifths by contrary motion are allowed if either of the chords or both 
 be one of the fundamental sevenths (10, 59). 
 
 2 The examples collected by Parry (48, 119 f.) that he dubs with the scornful title 
 "music-hall cadence" do not all deserve so severe condemnation when considered in this 
 connexion, whatever other faults they may exemplify. 
 
 3 Sacchi said consecutive fifths were admitted "(1) When the fifths lie between the 
 inner parts, not between the outer ones, (2) when the fifth with the bass is covered by 
 the sixth, (3) when the two fifths are so placed that the first ends one period, while the 
 second forms the beginning of another." Sacchi's explanation of his second exception 
 is excellent: "The fifth being here covered by the sixth, this consonance cannot be clearly
 
 xiv] PROHIBITIONS OF CONSECUTIVES 119 
 
 fusional series are put into operation against one another, both being 
 intervals characteristic of chords not a sort of repeated note, as the 
 octave may often be. The consonance makes too much (perpendicular) 
 unity or fusion, the dissonance too much (perpendicular) duality or 
 ruption for proper melodic flow. But their combination gives a new 
 balance of flow. Only, and that may well be a notable practical point, 
 the combination is most frequently such an interlocking of the two 
 elements as prevents either from standing forth and dominating the 
 progression. 
 
 This is exemplified in my sample. In 16 cases the fifth lay B-S; 
 of these only two are by contrary motion. The beneficial effect of 
 contrary motion is evidently not required for B-S. In 7 cases the position 
 of the fifths is A-T, and all of these are by similar motion. Ten cases 
 lie between the bass and the tenor or the voice next above the bass. 
 In this distribution we should expect the fifths to be more liable to fall 
 away from the rest of the chord, and so to become more than usually 
 noticeable. This view is perhaps supported by the fact that in three 
 of these cases contrary motion has been used, that in two the harmony 
 is of six parts, that one is produced merely by a sort of shake in the 
 tenor, and that in two one of the voices is helped to continuity by an 
 inserted passing note. Thus where there is special danger of the inter- 
 locking of the two opposed elements in the chords being lost, there 
 the composers have brought other compensatory influences to bear 
 upon the fifths. In one case S-A with similar motion the progression 
 is to a chord of the minor seventh, but support is given by special 
 melodic features, "the carrying out of a thoroughly established idea," 
 as Parry (45) says of this example. In one by Chopin the fifths occur 
 as a tremolo-like accompaniment to the bass figure. Five others are 
 by Dvorak, and one by Stainer. 
 
 Amongst the cases involving no discords the fifths are referable in 
 
 enough distinguished. The sixth is here much more noticeable than the fifth, because 
 it is the extreme parts that most strike the ear and draw the greatest attention. Besides, 
 the two neighbouring notes, the fifth and the sixth, form a dissonance, a so-called 
 acciacatura; and the effect of the acciacatura is a certain suspension and indecision of 
 sound, that makes us expect its resolution." 
 
 The admirable Sacchi, in fact, ends on an ultra-modern note by saying that we must 
 exercise moderation in judging of consecutive fifths, and that we must always consider, 
 besides, "the beauty and novelty of the thoughts, the regularity and artistry of the 
 progressions, the elegance and clearness of the melod}', the unity of the design, the force 
 of the expression," and the 'convenevolezza del costume' (the propriety of the feeling?). 
 In short we have to be equipped, not only with "the eyes of the face, but with those of 
 the mind" (57, si ., se i.). Plenty of scope for freedom and progress there !
 
 120 
 
 EXCEPTIONS TO THE 
 
 [CH. 
 
 ten cases to the use of passing, or more or less purely ornamental, notes. 
 They are all by similar motion. 
 
 The remainder include two by Mozart, one by Mendelssohn, one by 
 Rheinberger, the famous one from Beethoven's Pastoral Symphony, 
 three by Schumann (one between the beginning and end of phrases) 
 and one by Elert. Four of the preceding are by contrary motion. 
 Two (Prout and Gounod) are evidently intended to give the effect of 
 barbarous progression. 
 
 Among the non-discordant cases that by Karg Elert (23, 10) is not 
 only very beautiful, but at the same time unique in its build. It would 
 seem as if the two series of neutral intervals (sixths and tenths) were 
 able to outweigh that of the fifths : 
 
 KARG-ELERT'S EXAMPLE 
 Karg-Elert, " Ndher mein Gott " 
 
 Lent 
 
 9. 
 
 
 
 ' -^-r-^, 
 
 @ ^ 
 
 ~$~L A 
 
 
 
 
 
 ^^ ^~"^ \ 
 
 S *~ <TJ 
 
 1l$F4 
 
 d ' 
 
 
 
 T2 I" ^^ 
 
 
 
 
 
 
 
 
 * ^ 1 ^5 1 
 
 ij 
 
 
 
 
 F ^ 
 
 PPP 
 
 /w\ , 
 [(Ty * - c 
 
 i ' 
 
 s 
 
 
 ^^ 
 
 
 Sw. Salicional. 
 
 While progressions (involving fifths) between tonic and dominant 
 or subdominant of course occur, many other connexions appear .in the 
 most pleasing cases. From the cases I have collected I cannot persuade 
 myself that the harmonic connexion of chords (with the probable 
 exception of the one just mentioned) plays the important part in the 
 degree of acceptability of consecutives that Gladstone and Shinn 
 suggest. But I do not wish to put forward the groupings I have just 
 given as more than mere suggestions. The subject calls for an extensive 
 treatment on statistical (and experimental) lines, and for that the small 
 number of examples I have collected is only profitable in broadest 
 outline. In all probability they form a very special and biassed example. 
 But there is no doubt that the effect of the discords on the fifths stands 
 out prominently even so. And the features of the pleasing uses of fifths 
 they reveal, generally seem to be compatible with the theory of the
 
 xiv] PROHIBITIONS OF CONSECUTIVES 121 
 
 basis of the prohibition that was advanced in the preceding chapter. 
 They all either obliterate the special effect of fifths or they go to 
 strengthen what the fifths weaken the even flow of the different 
 voices. 
 
 For consecutive fourths with the bass it is stated by Shinn (58, 166) 
 that the prohibition does not apply to progressions in which either of 
 the chords forming the progression is a discord. Front, as we saw, 
 allowed them only " when the second of the two is a part of a fundamental 
 discord or a passing note" (cf. 58, 285 f.).
 
 CHAPTER XV 
 
 HIDDEN OCTAVES AND FIFTHS, ETC. 
 
 IN all treatises on harmony a regular counterpart to consecutive octaves 
 and fifths is found in the special treatment of so-called hidden octaves 
 and fifths. The traditional explanation of the latter sets them into 
 direct connexion with the former. Two voices that approach a consonance 
 by similar motion are supposed thereby virtually to present consecutives. 
 Only, as the tones leading to the consonance are not really sounded, 
 they were styled 'hidden' (cf. 55). The theory is ingenious in so far 
 as it thus leaves only one thing to be explained, the bad effect of 
 consecutives. But Shinn (59) is not far wrong in saying that this theory 
 "can only be regarded as an interesting tradition of the past, which, 
 in the present day no intelligent musician can pretend to believe." 
 We are the more relieved from discussing it, as it rests not on any facts, 
 but on a mere assumption, namely that the listener unconsciously fills 
 out these slurs. Some objective strength might be given to the position 
 by claiming that there is some sort of a real melodic slur involved in 
 the process ; but even that theory would hardly get beyond a preliminary 
 formulation. For this melodic slur would not bring the phenomenon 
 into relation with consecutives for which the presence or activity of a 
 melodic slur has never been assumed. 
 
 As we have learned in the preceding, our best method will be to 
 find and to describe the system of facts of which ' hidden ' consecutives 
 form a part and to try to draw from this system an interpretation 
 sufficient to describe, and in describing to explain, its members and 
 their relations. And as the objective facts summarised by exponents 
 of systematic harmony themselves suggest the procedure, we may 
 take our systematic table of consecutives as a model or ideal of discovery 
 in dealing with ' hidden ' intervals. As before we shall set out from the 
 formulations of E. Prout (52), which have been summarised in the 
 previous manner in Table III. 
 
 TABLE III 
 
 ' Exposed ' Intervals (52 28 ff.) 
 
 Showing relations between (1) the stringency of prohibition, (2) the grade of 
 fusion of any interval, and (3) the prominence of the voice-parts in which the interval 
 appears. 'Exposed' intervals (or 'hidden' octaves, fifths, etc.) are such as are
 
 CH. xv] HIDDEN OCTAVES AND FIFTHS, ETC. 
 
 123 
 
 approached by 'similar' motion in the two parts, i.e. both parts rise or fall (in 
 pitch) to the interval in question. 
 
 
 O. 
 
 5. 
 
 4. 
 
 Third or 
 sixth 
 
 T 
 
 Seventh 
 
 Second 
 or ninth 
 
 B-S 
 
 forb. 
 
 forb. 
 
 + 
 
 + 
 
 ? 
 
 
 
 ? 
 
 B-A 
 
 f. 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 B-T 
 
 f. 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 S-A 
 
 f. 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 S-T 
 
 f. 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 A-T 
 
 f. 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 forb. = forbidden with exceptions; f. = hardly forbidden; 
 + = allowed; ?= recommendation against. 
 
 The formulations embodied in the Table are not held to be exact laws, as it 
 were, but only to be properly representative of the trend of opinion on the subject 
 in its grading of stringency from point to point of the Table. 
 
 The rules upon which this Table is based are the following (52, 28 ff.) : 
 
 (1) Hidden octaves are forbidden between the extreme parts; except, first, 
 between primary chords in root-positions (i.e. with the roots in the bass), when 
 the bass must rise a fourth or fall a fifth, and the upper part must move by step; 
 second, when the second of the two chords is a second inversion 1 , the bass note being 
 either the tonic or dominant of the key; and 3rd, when the second chord is another 
 position of the first. 
 
 Hidden octaves are, however, allowed between any other of the parts than the 
 two extreme parts, with one important exception. It is strictly forbidden to move 
 from a seventh or ninth to an octave by similar motion between any two parts, 
 when one part moves a second, and the other a third. This is the very worst kind 
 of hidden octaves, and must be most carefully avoided. 
 
 (2) Hidden fifths are forbidden between extreme parts; except, first, in a pro- 
 gression between primary chords (tonic to dominant, or subdominant to tonic), 
 with the upper part, as with hidden octaves, moving by step. The first of the two 
 chords is not (as in the case of octaves) restricted to root-position (two examples 
 are given of it in first inversion); second, from the root position of the chord of 
 the supertonic, with the third in the upper part, to the chord of the dominant, 
 when the bass falls a fifth, and the upper part falls a third; and third, from one to 
 another position of the same chord, exactly as with hidden octaves. 
 
 1 Here there is sharp opposition with Tchaikovsky who says (76, so): "Concealed 
 octaves sound particularly unpleasant when the octave appears as doubled Fifth in the 
 six-four chord (or as doubled third in the sixth)." In both examples given by Prout 
 as good the octaves are given by a doubled fifth. This disagreement between the two 
 theorists is not due to confusion of the requirements of the hidden octave with those of 
 the fourth from the bass, but reflects their attitude towards the hidden octave only.
 
 124 HIDDEN OCTAVES AND FIFTHS, ETC. [CH. 
 
 (3) When two notes making a dissonance with one another (such as second, 
 seventh, or ninth) are taken without preparation that is, if neither of them has 
 been sounded in the same voice in the preceding chord it is better that they should 
 enter by contrary than by similar motion, especially in the extreme parts. 
 
 Tchaikovsky (76, 59 ff.) has treated hidden octaves and fifths in some 
 detail. Of the examples he gives as bad all would be forbidden by 
 Prout, except the progression to a six-four chord with doubled fifth 
 cited above; Tchaikovsky, besides, allows the octaves or fifths "when 
 they occur in the connexion of a triad with the Dominant chord, and 
 when the seventh is prepared in an inner voice," a case that Prout 
 does not mention. Tchaikovsky's classifications overlap one another 
 considerably, so that it is difficult to see clearly how far they coincide 
 logically with Prout's. They may be summarised as follows : 
 
 (1) Octaves and fifths in the outer voices are often very disagreeable 
 and should be avoided by all means : (a) when the upper voice proceeds 
 by a jump, but even then the progression is by no means unpleasant 
 in the connexion of a triad with the dominant chord when the seventh 
 is prepared in an inner voice; (b) when all the voices move in parallel 
 motion; (c) when the outer voices move in parallel jumps, even though 
 the inner voices remain stationary or progress stepwise. 
 
 Here there is evidently an opposition between jumps and steps or 
 stationariness : (a) one jump in the upper voice, (b) one jump worsened 
 by more parallel motion, (c) two jumps bettered by steps or stationariness. 
 Prout refers to step only in the upper voice, seeming thus to indicate 
 greater importance in the melodic continuity of that voice (even over 
 the bass) 1 . 
 
 (2) Concealed progressions between an outer and an inner voice 
 are disagreeable when the jump lies in an outer voice. (C. H. Kitson 
 30, so formulates a similar rule for three parts as against four parts, 
 for which it is not required.) 
 
 (3) Between the inner voices concealed octaves are "entirely out 
 of the question, arising solely in consequence of bad voice-leading"; 
 "concealed fifths, however, are permitted, provided the voice leading is 
 natural." (The word 'however' seems to imply that the octaves are 
 here meant to be more forbidden than the fifths.) 
 
 On the whole we may infer a grading of prohibitions, greatest for 
 B-S, least for A-T and medium for the intervening pairs (cf. 55), while 
 the minimum for the fifth in A-T is less than for the octave. Thus 
 we get from Tchaikovsky's analysis the same general result as from 
 
 1 But cf. below, p. 130, note 2.
 
 xv] HIDDEN OCTAVES AND FIFTHS, ETC. 125 
 
 Promt's. Of the examples given by Prout as good or bad all would be 
 similarly styled by Tchaikovsky except (1) the six-four case already 
 noted, (2) the progression to another position of the same chord, which 
 Tchaikovsky neither mentions nor exemplifies, and Prout's second excep- 
 tion to the rule against hidden fifths, which stands in systematic isolation 
 amongst Prout's other rules, as far as one can see. 
 
 Jadassohn (26, 36, 70) likewise forbids concealed octaves and fifths 
 when both voices leap, "no matter in what connexions and in what 
 direction, if in outer or inner voices, or one outer and one inner voice," 
 except when it is a case merely of inversion, or of interchange of voices. 
 One leap is not so bad, especially if it is not in the upper voice (Sopr.), 
 even in a progression between the triad on the mediant to that on the 
 submediant, or similarly from the submediant to the supertonic (26, 37). 
 There is some sign of greater freedom with the fifth than with the 
 octave : for "if the upper voice moves by a degree, and one of the lower 
 voices by a skip, the concealed fifth between all the voices is without 
 hesitation permitted, provided all the voices do not move in the same 
 direction 1 " (26, 72); in the case of octaves a slight reservation seems to 
 exist in the case of connecting the chords on the second and fifth 
 degrees (26, 37). In the chord of the dominant seventh a fifth may 
 occur even after similar motion of all the voices, but not along with 
 a diminished fifth. 
 
 The exceptional treatment of the progression from a chord on the 
 second to one on the fifth degree may have some connexion with Prout's 
 second exception to the rule for hidden fifths, but it seems to contradict 
 it rather than to support it. The systematic connexions of this case 
 are not at all evident. Jadassohn also cites the modifying effect of pitch- 
 direction on this succession: "Concealed octaves in the succession of 
 the chords on the second and fifth degrees, when downward, are not 
 faulty, as this connexion does not sound harsh" (26,35); they must, 
 however, be avoided in upward progression (26, 36). These points do 
 not affect the main result, but only point to other modifying factors, 
 which would each require a separate treatment, i.e. a search for their 
 systematic setting. 
 
 One other theorist is worthy of mention. Shinn (59) says : 
 
 Viewed from the modern standpoint, the term 'hidden' itself seems a mis- 
 nomer.... When so-called hidden octaves and fifths do produce an objectionable 
 musical effect, it is obviously due to the fact that the octave or the fifth which 
 ia present is itself thrown into undue prominence, and what is unsatisfactory in 
 
 1 Cf. below, p. 126.
 
 126 HIDDEN OCTAVES AND FIFTHS, ETC. [CH. 
 
 the progression is the result of the exceptional 'exposure' of the perfect interval 
 which is present, and not to a faulty progression which is supposed to exist in the 
 imagination of the listener. This fact is now admitted by the more progressive 
 amongst musical theorists, and in modern text-books the traditional and misleading 
 term 'hidden' is being gradually displaced by the more accurate one 'exposed,' 
 the universal adoption of which is certainly desirable in the interests of all students 
 and teachers of Harmony (cf. 58, aestr.). 
 
 It is disappointing to find so interesting a statement made without 
 any citation of the author of this change of interpretation. I have not 
 as yet met with the term 'exposed' in any other writer on harmony 
 except C. H. Kitson (30, 49) and prefer meanwhile to consider F. G. Shinn 
 himself to be the author of any distinct theory of 'exposure' in these 
 intervals, as he is certainly the only satisfactory exponent of the idea. 
 It is obvious that his interpretation is much more in line with the trend 
 of our own exposition than is the older term 'hidden,' although the 
 latter originated quite properly in an attempt to connect this set of 
 facts with that regarding consecutives. The mistake the traditional 
 theory made was to cling to the mere external form presented by the 
 consecutives as the essence of their offensiveness instead of getting 
 behind that form to the true essence. That essence, we have seen, is 
 most probably the break in the usual melodic connexion due to the 
 greatness of the consonance or dissonance in question and to its 
 ' exposure ' by similar motion. 
 
 Shinn then points out that " some of the most eminent continental 
 theorists impose restrictions upon the employment of octaves and 
 fifths so formed [exposed], either between two inner parts or between 
 one inner and one outside part, which are not recognised by English 
 theorists." He then cites Tchaikovsky and Jadassohn and remarks 
 in connexion with an example constructed to fit the rule quoted from 
 Jadassohn above (p. 125, note) : 
 
 When such restrictions and prohibitions extend so far as to describe as bad 
 and to forbid the employment of ["such a progression,"] it is doubtful whether 
 such rules are not merely devoid of all musical authority, but whether they possess 
 any value even for the purposes of mental discipline; whether, in fact, they do not 
 tend to make the introduction of any kind of spontaneous musical thought into 
 the work of the student absolutely impossible. 
 
 He himself then proceeds to consider their employment "from the 
 point of view adopted by English theorists, that is, when they are 
 formed between the outside parts"; and he suggests on the basis of 
 
 the examination of many examples of exposed octaves and exposed fifths... that 
 the effect of such progressions varies (and is more or less satisfactory) according
 
 xv] HIDDEN OCTAVES AND FIFTHS, ETC. 127 
 
 to the extent to which the particular effect of the exposed interval dominates the 
 effect of the second chord of the progression. When the nature of the individual 
 chords forming the progression, their harmonic relationship, the number of parts 
 employed, or their general progression, is such as to neutralise the effect of the 
 exposed interval, no unsatisfactory effect is produced. 
 
 By the nature of the individual chords is meant whether either or both of the 
 chords be a discord, or whether both are concords. When the second chord is a 
 discord, to whatever extent an exposed interval may be thrown into prominence, 
 the effect of this exposure is almost invariably neutralised by the dissonant character 
 of the chord. 
 
 Shinn gives two examples, one for the octave and the other for the 
 fifth, each occurring in a chord of the minor seventh (on supertonic and 
 on dominant). 
 
 He then discusses the effect of harmonic connexion upon c exposure ' : 
 
 In connexion with triads and their inversions whose roots are a fourth or a fifth 
 apart, and which therefore have one note in common, when one of the parts moves 
 by step, the other part leaping, the effect is rarely unsatisfactory. When both 
 parts leap, especially in a downward direction, neither part leaping more than a 
 fifth, the effect may be excellent. 
 
 When the roots of chords are a third or sixth apart, the effect depends 
 partly upon the degrees of the scale upon which the chords stand, 
 "some being bold and strong, others weak and unsatisfactory." Down- 
 ward approach is usually a favourable circumstance. The degrees used 
 also affect triads on adjacent notes. Inversion also : 
 
 When one of the chords is an inverted form, and the highest part moves by step, 
 while the lower part leaps either a fourth or a fifth, the effect is almost invariably 
 good.... An exposed octave formed between two such triads, the second being in 
 its second inversion, is also unobjectionable when one part moves by step and the 
 other leaps a fourth.... The strong and characteristic movement of the two parts, 
 by a second and by a fourth or fifth, exerts considerable influence in the direction 
 of strengthening the effect of the progression (cf. 58, 27211.). 
 
 Here we see a number of factors influencing the essential factor in 
 question the degree of consonance of the ('exposed') interval; we 
 can see to some extent the direction of the influence, whether it is 
 favourable or unfavourable and we can perhaps assess them against 
 one another. But the general result is hardly as clear as is desirable for 
 scientific aesthetic purposes. Finally, he points out that "progressions 
 which are unsatisfactory in two or even three parts may be quite good 
 in four or five parts." This he says is due partly to the doubling of 
 notes common to both chords, partly to the distribution of the listener's
 
 128 HIDDEN OCTAVES AND FIFTHS, ETC. [CH. 
 
 attention over a greater number, when "the progression of any two 
 parts (even when they are outside parts) must in some corresponding 
 proportion become less noticeable" (cf. 50, 255 f., 296). 
 
 The result of this analysis of representative authors may seem 
 hopelessly confused, as it must certainly be highly unsatisfactory from 
 any practical point of view 1 . But we can skim the common general 
 effect off the differences peculiar to the different writers. (1) The 
 objectionable phenomenon in question is presented by the octave and 
 the fifth (all theorists agree), and somewhat more strongly by the 
 octave than by the fifth (Prout, Tchaikovsky, Jadassohn and Shinn) 2 . 
 But whereas in the case of consecutives the objectionable feature 
 appeared regularly and was only mitigated or made tolerable by contrary 
 motion, in this case (2) it does not appear until it is 'exposed' by the 
 unfavourable effect of similar motion. Parallel motion in other voices 
 than those primarily concerned increases the unfavourable effect (T., J.). 
 (3) The effect is worst in the outer voices (P., T., S.), it is least in the 
 inner voices (T.) and of medium degree between an outer and an inner 
 voice (T.). 
 
 The unconcern of the English exponents to any but the outer voices 
 is, of course, not inconsistent with a gradation of the acceptable effect 
 in the rest of the series of voice pairs. Differences of opinion on the 
 position of the point of change from desirable to undesirable effect are 
 inevitable in the midst of so many fluctuating factors. It would be a 
 mistake to wish for that reason to throw aside all the formulations of 
 previous exponents, even if we feel that their terms do not properly 
 express our judgments, and to start a search for an entirely new set 
 of formulative notions. A prejudice of that kind may make success 
 impossible. We should keep our attention chiefly directed upon the 
 systematic trend of the rules of previous analysts, in case our difference 
 from them may simply be settled by a shifting of the border line of 
 acceptability without any radical change of the basis of judgment. 
 
 This counsel holds not only for the primary features of the situation, 
 but also for the minor factors which create exceptions from the main 
 rules. 
 
 Thus (4) the progression by step has a favourable effect, while leaps 
 
 1 Kitson says (30, 49): "It is impossible to find any basis of general agreement as to 
 which exposed consecutives are objectionable and which not." 
 
 2 Shinn says "the effect of exposed octaves distinctly differs from, and is often far 
 less satisfactory than that of exposed fifths" (58, 271).
 
 xv] HIDDEN OCTAVES AND FIFTHS, ETC. 129 
 
 are unfavourable. Two jumps one in each voice are worse than one, 
 while the effect of any one jump varies with the prominence, especially 
 the melodic prominence, of the voice that bears it. A step accordingly 
 is more powerful in the highest voice 1 , then in the lowest, etc. Jadassohn 
 says jumps in any two voices are bad, while one is least bad if it is not 
 in the upper voice. Tchaikovsky deems two jumps in the outer voices 
 very bad, one jump worse in an outer than in an inner voice, and not 
 usually admissible in the highest voice. Prout requires a step in the 
 upper voice to support the favourable influence of close relationship 
 of chords. And Shinn thinks the strongest effect is produced when the 
 upper part moves by step (58. 272 ff.) 2 . He also desires two jumps (in 
 outer voices, of course), to be mitigated by downward motion and 
 restriction of the leap to the 'emmelic' range (not more than a fifth). 
 Melodic continuity, then, counteracts the bad effect of the consonance 
 exposed by similar motion. This agrees with our conclusion for con- 
 sec utives, that their offence was a breach of the required melodic 
 continuity. (Two steps would, of course, often convert the problem 
 into one of consecutives. A tritone may be followed by a perfect fifth 
 if the lower part rises a semitone; then there are two 'steps.' Cf. p. 104 
 above, Rule II.) But while there is thus agreement in the general 
 trend, there is considerable disagreement between writers as to the 
 margin of pleasant effect. But the drawing of these border lines does 
 not primarily concern us here. 
 
 (5) Writers (T., J., S.) agree that the bad effect is covered over 
 again in characteristic discords, e.g. that of the minor seventh. One 
 might argue that the exposure should then be greater as the two intervals 
 stand at the ends of the fusional scale. But the direction of pull, as it 
 were, is opposite in the two cases : one is consonance or unity, the 
 other is dissonance or duality. Perhaps this combination achieves 
 something like the neutrality of the thirds and sixths (without, of 
 course, rendering the chord consonant as a whole). 
 
 (6) 'Connexions' between the chords or the tones seem favourable. 
 When the chords only differ in position, the bad effect is annulled, 
 probably because melodic connexion is readily attained from one 
 
 1 Which in modern music is usually not only generally, but also thematically, melodic. 
 Cf. Tovey's definition of melody as "the surface of music." 
 
 2 In the case of the tritone followed by the fifth, already referred to (p. 1 04), the progression 
 is satisfactory in so far as the leading tone, which forms the lower note of the tritone, 
 moves by semitone to the tonic. But if this movement is exposed on the bass, it so strongly 
 'announces* the coming fifth that we get a specially exposed fifth, or else the leading tone 
 refuses its chief function by falling (cf. 52, n, 101). 
 
 w. F. M. 9
 
 * 
 130 HIDDEN OCTAVES AND FIFTHS, ETC. [CH. 
 
 position to another of the same chord or pattern. Here the second 
 tonal mass has been facilitated for the mind by the occurrence of the 
 first one, so similar to it. Thus the melodic connexions are favoured. 
 Similarly when the bass moves from one important, and therefore 
 familiar (or easy), point of the tonal system to another (tonic to dominant 
 and vice versa, subdominant to tonic and vice versa], or when it merely 
 moves by similar intervals, fourth and fifth), continuity in the bass 
 is made easy; and if step relieves any difficulty in the upper voice, the 
 effect should be excellent 1 . But there seems no strong reason why 
 with Shinn these relations should not be inverted to step in bass and 
 easy leap (e.g. fourth or fifth in upper voice). Only, the system of grada- 
 tions we have shown would incline us to expect not quite so good an 
 effect in the latter way as in the former 2 . But, of course, both ways 
 may be acceptable. The degrees of the scale used, other than tonic, 
 dominant, and subdominant, may further modify the ease of progression 
 according to their familiarity in the tonal system. 
 
 (7) Prout's note regarding the seventh or ninth before the octave 
 seems clear, because these intervals, even in isolation, strongly suggest 
 the unity and repose of the octave. This suggestion is probably even 
 increased by the stepwise movement of one of the voices, which will 
 ordinarily be accompanied by the movement of a third in the other 
 voice, when the following interval is an octave. Thus the octave about 
 to be heard will be strongly suggested, and so will be more exposed 
 than ever. 
 
 (8) In dealing with consecutives we have already encountered the 
 effect of a larger number of parts than four. Prout (52, 305) says of 
 this : " In proportion as the number of parts, and therefore the difficulty, 
 increases, the stringency of the rules relaxes. These hidden fifths and 
 
 1 These points are nearly all applicable to the unison, which may not generally be 
 taken by similar motion, unless in the progression from dominant to tonic, with the help 
 of step in one voice also perhaps (cf. 52, 3if.). The voices in which the unison appears 
 will generally be neighbouring ones, of course. The unison by similar motion seems to 
 be more strictly forbidden than is even the octave. The unison, of course, gives the 
 clearest expression to the characteristic of consonances their apparent unitariness. 
 
 2 This would imply that the leap of a fourth or fifth is a more potent factor than is 
 a step. For, as the bass gives greater exposure than the soprano, x (fourth or fifth in bass) 
 
 +y (step in soprano) would be greater than y (step in bass)+x. This inference might 
 be preferable to that suggested on page 124 above (greater melodic prominence of the 
 soprano), which is irreconcilable with the general predominance of the bass except in 
 so far as the soprano in modern music is often thematically the most melodic voice, though 
 it can never be generally or in its mere sonorousness the most melodic. Cf. the later 
 discussion on the most general aspect of melody.
 
 xv] HIDDEN OCTAVES AND FIFTHS, ETC. 131 
 
 octaves are allowed, even when both voices leap; consecutive octaves 
 and fifths by contrary motion may be used freely; we even meet in the 
 works of the great masters with examples of a doubled leading note, 
 though it is better to avoid this, if possible." It would probably be 
 wrong to suppose that the mere difficulty of the work is the basis of 
 license, as if an ugliness could be compensated by such an extraneous 
 reason. It seems preferable to suppose that the succession is admitted 
 because the many parts steady each other and maintain a general 
 distribution of effects. The obscurity of the inner parts in four part 
 writing has the same origin. 
 
 We shall have some opportunity later of considering the influence 
 of inversions. And the difference between upward and downward 
 motion may be neglected for the present; it is not quite clear what the 
 relation of this difference is to the matters referred to in chapter vin. 
 The implication is that descent strengthens melodic continuity, possibly 
 because descent has more of the character of a return to the starting- 
 point, while ascent is departure (cf. above, p. 52 f.). Prout's unique 
 rule regarding the supertonic chord is probably more or less of an 
 accident in his usually so systematic work. It seems a composite of 
 downward motion and of progression of a fifth in bass. Only the latter 
 factor is invoked by Shinn to explain the case (58, 49). 
 
 In conclusion we may say that the Table at the beginning of this 
 chapter, based upon Prout's formulations, properly represents the 
 trend of differences it refers to, and may therefore be taken as valid 
 material upon which a theory of the phenomenon in question may be 
 raised. As we have so often suggested, the exact place at which one 
 draws the line between what is desirable and what not, depends on 
 variable subjective factors. But the scale of differences which make 
 different aesthetic reactions possible is objective and invariable, as is 
 also the general trend of their influence upon the aesthetic reactions 
 of an individual. The aesthetic realm is not to be considered the sport 
 of caprice because there is no disputing about tastes. The latter state- 
 ment is true only because one man may carry subjective inclinations 
 and influences about with him that another does not possess and finds 
 of no particular interest. If these so affect him as to make his aesthetic 
 judgments contrary to another's, the latter cannot offer to dispute him 
 out of their influence. Between one man and another only those forces 
 are subject to common analysis or discussion that are objective to 
 both of them, or that are primarily rooted in the artistic work itself. 
 
 92
 
 132 HIDDEN OCTAVES AND FIFTHS, ETC. [CH. xv 
 
 A study of the factors that operate in some individuals and not in 
 others is work for a psychology of personal differences, not for a science 
 of the constitution and potencies of the aesthetic objects themselves. 
 About these and the laws of their being dispute is as little excluded as 
 it is in dealing with nature itself. 
 
 It is often said that artistic rules are hardly formed before they are 
 swept away into oblivion by the stroke of some genius. That and all 
 such expressions are radically wrong, as wrong as it would be to say 
 that no sooner is a law of nature discovered than some engineer sweeps 
 it away by showing how to circumvent it or (apparently) to oppose 
 and to reverse its action. The will of a person may be overruled and 
 forgotten, but no one supposes the art of a past century to have been 
 a tyrant's will. A genius breaks no rule of art. He only fulfils it the 
 more by finding influences which unite with it to produce effects it 
 would be incapable of producing alone. After all no one really believes 
 this fable of the genius. You always have to be the genius before you 
 can have his power to make rules disappear. You must have his know- 
 ledge and experience. In fact, you must know 1 how to do it. And it 
 is right that the beginner should learn the big facts and rules first, as 
 really and permanently valid, not irreconcilably valid or as pedants' 
 foibles to be discarded later. He must first learn the broad effects and 
 then progress towards the subtle ones; and he will do so easily and 
 willingly when his discipline can be set before him in systematic form. 
 When its foundations have been well expounded, he will be able of 
 himself to carry them forward into their interactions with one another 
 much more readily and steadily than he would if he had to learn them 
 all separately and unintelligently. 
 
 1 Or 'feel' (in sensory constructions). A composer who does so, may perhaps not also 
 know the method or law inherent in his feeling. It is for the theorist to find that out, if 
 the composer does not first discover it himself. In his " Philosophy of Modernism (in 
 its connexion with music)" Cyril Scott says: "It is a fact, almost a truism, among 
 enlightened musicians, that we learn the rules only in order to know how to break them; 
 but the real quarrel arises in how often and how far one is permitted to break them. 
 The truth is, in reality there are no rules. There are merely conventions; and these 
 conventions have altered with the advent of each new master" (p. 27 f.). 'Breaking 
 them' suggests the overcoming of law by its better fulfilment; but, of course, there is 
 also such a thing as the mere ignoring of rules.
 
 CHAPTER XVI 
 
 A FOURTH FROM THE BASS 
 
 IT is a notable fact, which was emphasised in the preceding chapter, 
 that similar motion is the primary condition of the prohibited 'exposure ' 
 of the two intervals octave and fifth. The next most important condition, 
 embodied in the Table on page 123, is the prominence or exposure of 
 the pair of voices that bear the interval. We noticed that amongst 
 English theorists the prohibition of exposure extends only to the most 
 prominent pair bass and soprano. The intervals prohibited are 
 familiarly octave and fifth only ; but it is proper to place the dissonances 
 at the other extreme of the same Table because of the common desire 
 that they should not be exposed by similar motion. 
 
 We have thus detected at least a part of a system of facts that 
 forms a proper counterpart to the system obtained for consecutives. 
 We might offer to pass the rest of the system as the counterpart of 
 successive thirds and sixths the neutral region for which no prohibition 
 exists. The only difference would be that this neutral region is now 
 wider. And that would be in natural agreement with the greatly 
 reduced degree of prohibition set upon 'exposed' intervals. We might 
 even infer from the scope of this widening that the fourth as a consonance 
 borders more closely upon the neutral region than its common ranking 
 as a perfect consonance would suggest. That would also agree with 
 the system of consecutives; for in it we saw that only fourths from the 
 bass are forbidden. Or we might say that the fourth lies on the border 
 between high grade consonance and neutral sonance. 
 
 But we cannot accept any such simple solution without careful 
 inquiry. For it is a familiar fact of harmony that a fourth from the 
 bass must be used with great care. Many rules for its use have been 
 formulated and these must be carefully analysed before we can judge 
 accurately of the status of the fourth as an ' exposible ' interval. 
 
 We shall again base our summary in the first instance upon the 
 formulations of E. Prout (52, 66 fi.). 
 
 "Though it is possible," Prout says, "to take any triad in its second 
 inversion, the employment of any but primary triads in this position 
 is extremely rare." Macfarren (35, 68 ff.) had asserted that only those
 
 134 A FOURTH FROM THE BASS [CH. 
 
 on the tonic, subdominant, and dominant were admitted. It is, of 
 course, the ' root ' of the chord that is supposed to stand on these degrees ; 
 the lowest tone of the second inversion is a fourth lower. For purposes 
 of description these two authorities may be said to agree. The six- 
 four chord has not only to be approached, but also to be quitted, in 
 certain ways. 
 
 It may be approached : (1) either by leap or by step from the root 
 position of another chord; (2) by leap from another position of the 
 same chord ; (3) by step (but not by leap) from the inversion of another 
 chord; (4) from a different chord upon the same bass note. These 
 steps and leaps all refer to the bass note upon which the fourth stands. 
 
 It may be left : (1) by step of a tone or a semitone, upwards or down- 
 wards, and the following chord may be either in root position or in an 
 inversion; (2) by the same bass note or its octave bearing another 
 chord, provided the six-four chord is a tonic or a subdominant chord. 
 (This is the 'cadential six-four.') And in this case the six-four must 
 be on a strong accent, unless it has also been preceded by a chord on 
 the same bass note. (Evidently this cadential effect requires a sort of 
 rhythmical exposure of the six-four chord to create a ready disposition 
 for it in the listener's mind.) (3) By leap to another note of the same 
 chord (without change of harmony), provided that, when the harmony 
 changes, it returns either to its former note, or to the note next above 
 it or below it. In other cases it is not good for it to leap (32, 70f.). 
 
 Now we know already that for melodic continuity a leap is less 
 favourable than a step, and the repetition of the same note is easier 
 to follow than is a step. By abstracting this term from the above 
 rules and by grading those that are left over against it, we may attempt 
 to gauge the effect of the latter. By this process Table IV has been 
 constructed. In order to enter a single value at each point I adopted 
 as indicator the extreme permitted under each head. Thus 'by leap or 
 by step' means 'even by leap'; if a leap is permitted, so is a step, 
 of course. Similarly 'inversion' means 'either inversion or a more 
 favourable form root position.' But 'root position' means 'only root, 
 not inversion.' 
 
 The Table, then, indicates that the melodic continuity and ease of 
 the bass voice in a six-four chord are a matter of special concern. For 
 when favourable conditions are secured for it by conjunct motion or 
 by no change at all, the other parts have complete freedom of movement ; 
 but when its movement is made less cogent by a leap, the others should 
 be more favourable, giving at least either the same chord or (in
 
 xvi] 
 
 A FOURTH FROM THE BASS 
 
 135 
 
 approaching the six-four) proceeding from the clear cut stability of a 
 root position. 
 
 TABLE IV 
 
 Scheme of rules (after E. Prout, 52, 70f .) for approaching and quitting a 
 
 six-four chord. 
 
 
 Approaching a | 
 
 Quitting a | 
 
 Movement of 
 the Bass 
 
 From / i \ 
 chord \\) 
 
 In- /A\ 
 position \ 1 J 
 
 To-/f\ 
 chord V 1 j 
 
 In 
 
 position 
 
 By leap 
 
 Same ) | 
 Different j | 
 
 1 j Inversion 
 I | root 
 
 Same (1) 
 
 Inversion 
 
 By step 
 
 Different 
 
 Inversion 
 
 Different (3) 
 
 Inversion 
 
 None (or octave) 
 
 Different 
 
 Inversion 
 
 Different (2) 
 
 Inversion 
 
 The arrow-heads point in the direction of greater melodic continuity and ease. Each 
 entry in the Table indicates the maximum allowed. 
 
 Note 1. Provided that when the harmony does change, it returns either to its former 
 note or to the note next above it or below it (or to a note to which a correct progression 
 from the six-four chord could have been made, 58, ei). 
 
 Not* 2. Provided the six -four chord is a tonic or a dominant chord. And in this 
 case the six-four must be on a strong accent, unless it has also been preceded by a chord 
 on the same bass note (cadential six-four). 
 
 Note 3. In this case the six-four should occur on the unaccented part of the measure 
 (76, wf.). If the one is followed by another six-four, the latter will be cadential, and so 
 a rule of frequency for successive six-fours may be formulated (cf. 38, so). 
 
 The rules given by Prout thus evidently form a consistent whole, 
 which is at the same time thoroughly representative of the generalisations 
 of other writers on harmony. Minds may, of course, as we have already 
 noted, vary for subjective reasons in the exact margin of desirability 
 they draw. For a study of the effect of objective differences that concerns 
 us now, the essential consideration is the trend of the changes in 
 desirability. The smaller the range of personal differences, the more 
 reliable will this trend be. And here the range seems to be very small 
 indeed. 
 
 There is, then, in the six-four chord a sort of opposition or rivalry 
 between the two parts of the chord, the bass and the other parts. 
 The problem is : what is the basis and origin of this melodic rivalry? 
 
 A strong indication is given by the fact that the essential charac- 
 teristic of the chord is its fourth from the bass. This has been condensed
 
 136 A FOURTH FROM THE BASS [CH. 
 
 into the radical classification of the fourth as a dissonance. Such a 
 theory is, of course, erroneously extreme : the fourth cannot be set 
 down generically as a dissonance, for apart from the bass it is not at 
 all dissonant. But the indication at least requires us to consider whether 
 any special treatment has to be given to the upper note of the fourth. 
 
 Prout makes no statement on this point whatever, beyond pointing out 
 that the upper is the dissonant note and therefore should not be doubled 
 (52, 71). Macfarren (35, 90) pointed out that "the consonance of the 
 fourth in these three inversions [on tonic, subdominant, and dominant], 
 is proved by the entirely free progression of the 4th it comprises, which 
 is the assumed dissonant note of the disputants." Shinn (58, 61) says the 
 rules for six-fours "refer entirely to the progression of the bass part." 
 On the contrary Tchaikovsky says that the six-four chord "must in 
 any [other?] case be connected with one of the neighbouring chords 
 by means of its fourth; and into and from its other neighbour the 
 fourth must progress stepwise" (76,56). Mansfield (38,38) is more 
 comprehensive : "The 4th ... being a dissonant interval should be 
 approached and quitted conjunctly, and, if possible, in contrary motion 
 with the bass. Failing this, it should be prepared, i.e. heard as a con- 
 sonant note (or as an essential note, i.e. a note without which a chord 
 would be incomplete, such as a 7th in the chords of the dominant and 
 diminished 7ths) in the same part in the previous chord. If approached 
 by skip, it should be in contrary motion with the bass, and it should 
 be quitted by oblique motion with the bass when contrary motion is 
 not possible. These rules apply to almost all dissonant notes." Table IV 
 shows that oblique movement is frequently implied in the rules, e.g. over 
 against 'leap' and 'none.' Opportunities for similar motion would 
 arise mainly where the bass moved by step. Whether the number of 
 times it actually occurs is so small as to make a prohibitory rule useful 
 I do not know. One of Shinn's examples (58, 60, e) shows similar motion. 
 
 In favour of the consonance of the fourth there are to be urged 
 the work above expounded regarding fusion, a large part of all musical 
 experiences relating to the fourth (i.e. all apart from the bass), its 
 graded position in the Table of consecutives, and the beneficial effect 
 of (fundamental) discords upon consecutive fourths from the bass. 
 The beneficial effect of contrary motion claimed by Mansfield would 
 also correspond to its ranking as an 'exposed' interval next to the 
 fifth. There is no doubt that the fourth is the consonance of third grade 
 and that the fourth from the bass is to some extent at least an exposed 
 interval.
 
 xvi] A FOURTH FROM THE BASS 137 
 
 On the other hand the rules for its use seem to be far more numerous 
 and stringent than we might have expected from its grading after the 
 fifth. Contrary motion does not seem to be enough to reduce the 
 'exposure' without the help of conjunct motion in the lower voice 
 or even in both. And if consecutive fourths from the bass are condoned 
 in discords, we should expect to find a single fourth from the bass 
 a fortiori tolerable in a discord. But Prout, for example, says (52, KM) 
 that "the rules for approaching and quitting a second inversion apply 
 to the second inversions of discords as well as of concords." 
 
 This last statement seems to dispose, not only of the bass fourth 
 as an ' exposed ' interval, but as a dissonance as well. Why in approaching 
 and leaving an interval that is merely supposed to be a dissonance 
 should we have to take so much more care than in dealing with intervals 
 that are undoubtedly dissonant? If the rules are too complex to allow 
 us to assume the consonance of the bass fourth, they are just as excessive 
 for its dissonance. The characteristic of dissonances that appears in 
 the fourth is its tendency to suggest the major (or minor) third; it seems 
 to call for a resolution into that interval and to urge the whole chord 
 in which it appears in that direction. But, in contrast to regular 
 dissonance, we find that this characteristic can be suppressed by 
 appropriate circumstances. The discordant feature can be eliminated. 
 In no regular dissonance do we find that any method of approach or 
 departure from the interval will make it appear consonant, however 
 it may alter the trend of its impulse to resolution, or facilitate its 
 appearance in the music. 
 
 A trend of resolution appears most strongly in the 'cadential six- 
 four chord,' which is subjected to a rhythmical 'exposure.' The interval 
 is not thereby rendered dissonant; it stands forth clearly as a fourth; 
 we may even suppose that its grading as a consonance emphasises this 
 exposure to some extent; we have no reason to argue that in that case 
 it should be approached by contrary motion, which would reduce the 
 'exposure'; for its exposure is just what we desire at the moment, 
 seeing that we also give it a rhythmical exposure. But there are only 
 two degrees of the scale that properly invite this cadential resolution 
 the dominant and the tonic (tonic and subdominant chords). These 
 are the only degrees of the diatonic scale that have a semitone imme- 
 diately below them. The one gives a cadence on to the dominant, 
 the other on to the tonic. A cadence upon the subdominant is spoilt 
 by the tritone that there appears instead of the fourth. Where tonality 
 is well marked, the dissonantal tendency of the fourth is most useful.
 
 138 A FOURTH FROM THE BASS [CH. 
 
 But as it is not desirable to pursue a cadence of this kind whenever 
 a six-four chord is used, means have to be found to avoid the effect. 
 This is done by placing the chord upon an unaccented part of the 
 measure and by leaving it by step in the bass. The other procedure, 
 noted in Table IV, note 1, is not really a variant upon these two. It 
 represents only a temporary movement from the six-four bass. 
 
 The bass note of the fourth is, therefore, the most important of the 
 chord, in so far as progression from it is concerned. If it is exposed 
 (1) by being the bass note, (2) by being rhythmically accented, (3) by 
 being the tonic or dominant degree of the scale, it will produce the effect 
 of arrest of motion and strongly suggest the interval of the major 
 third that is so near to it, and so produce the cadence. If such a cadence 
 is not desired, the exposing conditions (3) may, and (2) must, be avoided, 
 and the remaining tendency to revive the cadential tendency must be 
 suppressed by giving the bass a strong melodic force progression by 
 step. 
 
 It is by reference to this tendency towards the third that we must 
 explain the special precautions to be taken in approaching the six-four 
 chord. Even when the bass fourth is skilfully introduced, it still tends 
 more or less strongly to suggest the third. All the more, then, should 
 we expect to find a tendency to confusion of melodic attachments 
 inherent in the approach to a bass fourth. Unless special care is taken 
 the melodic streams would tend to fall towards the third and a jar 
 of surprise would be caused by the fourth actually given. If this jar 
 is to be avoided, we must lead a strong melodic current upon the tones 
 of the bass fourth and especially upon the lower or bass tone which 
 gives the pitch or centre to the whole chord. 
 
 Thus we see that the problem of the fourth is akin to that of 
 consecutives. The difficulty in both is the maintenance of clear, un- 
 ambiguous melodic lines. But they are otherwise different. In con- 
 secutives and exposed intervals the obstruction is chiefly caused by 
 the pronounced consonance or dissonance inherent in the single interval 
 itself. In the fourth it is due to the proximity of the fourth to the 
 interval of the third. The distracting influence is external. How it 
 comes to have this effect we shall consider later on. 
 
 These difficulties and uncertainties into which we have been led, 
 show us more than ever how desirable it is that information should 
 be gathered about the treatment, not only of consecutive intervals, 
 but also of all single intervals in relation to the circumstances under
 
 xvi] A FOURTH FROM THE BASS 139 
 
 which they are used, on a statistical basis. To many the statistical 
 method may seem to be a dry as dust business. But if we are to over- 
 come the divergence of opinion which characterises the exposition of 
 harmony and to obtain a body of definite, generally accepted knowledge, 
 it is our only hope. No other method, either, will ever permit us to 
 subtract from the treatment accorded to an interval the part that is 
 probably due to a certain influence so as to leave us with the amount 
 due to any other. These apportionings call for a quantitative treatment, 
 and that is procurable only by statistical methods. At the present 
 time every writer who works out the rules of harmony for himself 
 has to do over again work done by many others before him. And he 
 does not relieve his successors of the necessity of repeating the work. 
 At best he can cover only a small range of the task, and in doing so he 
 is liable to be greatly influenced not only by the generalisations of his 
 predecessors but also by his own special preferences and prejudices. 
 With statistical methods, however, a piece of work, if it is once done 
 thoroughly, is not only finished, but is open to the view of every one 
 else. It need be repeated at most only once for the purpose of verifica- 
 tion. Anyone who desires to continue research of a problem already 
 investigated, will have an opportunity in testing the validity of statistics 
 already derived from one composer or period with those to be derived 
 from another. By this means a reliable history of the developments of 
 harmony would in time result. Once the methods of this kind of 
 statistical research were well known, such repetitive tasks might be 
 given to younger students who desire to follow the work of any composer 
 with close analytic attention. As things are at present such a historical 
 view is only present in feeble outline of the broadest kind. We do not 
 know even whether the formulations of the best analysts are complete 
 or how far they still fall short of approximate completeness. 
 
 The results we have obtained thus far alone suffice to convince us 
 that with the proper systematic approach and outlook there is every 
 prospect that the science of harmony will one day attain a high grade 
 of precision, if statistical methods are carefully pursued. It will become 
 capable of systematic treatment that should make its apprehension 
 easy and comprehensive. This possibility seems much more probable 
 for music than for the pictorial arts. Music has the advantage of 
 operating with units of sound that are capable of only slight fluctuations 
 from certain forms their pitches. There are perhaps many people 
 who would look upon such an accurate science of divinae musicae as 
 a disastrous calamity. But that is really an absurd point of view. An
 
 140 A FOURTH FROM THE BASS [OH. xvi 
 
 art can only be furthered by a greater knowledge of its essential nature. 
 Its progress should then be more rapid and sure. We could estimate 
 the possible lines of advance with great probability of success, and if 
 no new vistas seemed likely to open up along our present lines of progress, 
 those who are in search of new lands would know to what point of 
 the system of sounds that leads to music they would have to recede 
 in order to be able to diverge upon strange paths of new outlook. And 
 that would be a great gain. We do not know whether much of the 
 experimentation in music of to-day is not from the outset a waste of 
 time. With a science of music well developed we should be able to 
 judge on this matter with some certainty. The world is not the greater 
 or freer to a genius for his ignorance. He does not create by personal 
 decree, but by discovery of new effects which were already laid down 
 as possibilities in the systematic growth of the art before he took it 
 over. And all unaided discovery is slow and painful, even to a genius. 
 With knowledge discovery may become possible to many others besides 
 the genius, who may then climb nearer to his summits. 
 
 In bur knowledge of the physical basis of pitch we have a very 
 accurate science of the fundament of music. Here our knowledge is 
 practically complete. But no one supposes that the divine art has 
 become any more earthly for that reason. Why should a science of the 
 art itself degrade it any the more? Those in whom knowledge and its 
 precision tend to dispel the attractions of beautiful and wonderful 
 things 1 will still be able to keep their minds unsullied, if they so desire. 
 But the charm of mystery does not lie in any vagueness of the sensory 
 stuff of art or of its beauty, but in all the longing hopes these finished 
 forms arouse in our minds. We feel the course of life as it might be, 
 were we not our own poor guides stumbling towards ends we can only 
 dimly discern, but the stuff upon which some divine artist had chosen 
 to lay his wondrous hand. We move in the divine thought wrapt up 
 in that stuff of sound and we long to have and to be its life. 
 
 Nevertheless the sounds we hear have the precise structure of 
 crystal and their beauty is a chiselled gem. Their sciences may be their 
 equal and counterpart. 
 
 1 Cf. A. E. Hull, Cyril Scott, London 1918, p. 78 f . : " Like Debussy, he [Cyril Scott] 
 would protest against the dissection of his music^ as if it were a piece of curious clock- 
 work mechanism. In the Revue Blanche in 1891 the French master wrote, "As children 
 we were taught to regard the dismemberment of our playthings and toys as a crime of 
 high treason, but these older children still persist in poking their noses where they are 
 not wanted, endeavouring to explain and dissect everything in a cold-blooded way, thus 
 putting an end to all mystery."
 
 CHAPTER XVII 
 
 COMMON CHORDS OR CONCORDANCE 
 
 FROM chapter xi till now we have been engaged essentially in the study 
 of single intervals of two tones. It is true we have considered them 
 generally as they stand in harmony of four or more parts. But our 
 interest centred primarily in the interval of two tones itself, as if it 
 were the element of structure of four part harmony. The results of 
 our study enable us now to show the exact manner in which intervals 
 generally follow one another in harmony of two or more parts. 
 
 These results seem both to enrich and to modify the outlook afforded 
 by such previous knowledge as had been systematically sifted. That 
 culminated in the notion of a fusion inherent in each pair of tones 
 themselves and not borrowed from any of their adjuncts. Consonance 
 and dissonance were the opposite poles of this fusion. And consonance 
 seemed obviously to be the ground upon which the pleasures of music 
 mainly stand, although they were evidently greatly enhanced by contrast 
 with dissonance. A general statement of this kind, however, seemed 
 plainly unable to give any sort of adequate expression to the whole 
 nature of music in many parts. The theory showed a crudity and 
 insufficiency very like that of the primitive music of the discantors 
 in comparison with the modern art. 
 
 The outlook presented by the concept of fusion was clouded by the 
 emphasis laid upon the approximation of the high grade fusions to the 
 unity and balance of a single pure tone, and the ensuing tendency to 
 carry that notion over into the general idea of consonance and dissonance 
 as they enter into modern music. Thus the function expected from 
 an octave or a fifth in music was such as would express its unity or 
 approximation to the balance of a single tone. The function of a second 
 or a seventh would reveal its tonal duality. But the merest glance 
 at the nature of music seemed to contradict any such conclusion. For 
 in the prototype of musical groupings of tones in the common chords 
 we find three essential tones and three intervals. Approximation to 
 the balance of a single tone is out of the question. Even a dull ear 
 would detect plurality in every case. And the interval of the chord 
 most essential to its musical functions is not, as the theory of fusion 
 would most likely suggest, its fifth, but its lower third. But it was not
 
 142 COMMON CHORDS OR CONCORDANCE [OH. 
 
 apparent from the previous theory why the lower third should be more 
 important than the upper one. In fact attempts were made to explain 
 the differences of major and minor triads in terms of the reversal of 
 the positions which the thirds occupy in them (43; cf. 60, 35 f., 44-53 
 (Zarlino), 219 ff. (Rameau), 293 ff. (Tartini), 367 ff. (Hauptmann), 385 ff. 
 (Oettingen), 387 ff. (Riemann)). But without avail (cf. 67, 84ff.; 71,333; 
 79). We do not look upon the one chord from below, and upon the other 
 from above. 
 
 The great importance of the thirds in modern music did not seem 
 to fit into the theory of fusion. For they were neither high grade 
 consonances nor high grade dissonances. A means of reaching their 
 musical function seemed indeed to ensue upon the distinction of grades 
 of pleasantness in intervals. Thirds and sixths rank high in the scale 
 (29, 194). But it must be evident that mere pleasantness without a 
 justifiable basis of pleasure is a weak reed for any theory of music to 
 lean upon (cf. 71, 351 ff.). 
 
 So the outlook upon music seemed to be blocked completely. There 
 seemed to be no means of approach to music as we find it. And it was 
 inevitable that in time an attempt should be made to make a new 
 start, to find a new notion upon which the functions of chords might 
 be grounded. 
 
 This idea Stumpf attempted to supply in his notion of concordance 
 as distinguished from consonance (71). 
 
 Two notes are consonant when they sound together so as to fuse 
 into an approximate unity, whether the component tones are distin- 
 guished or recognised at the same time or not. The greatest degree of 
 unification appears in the octave. It lessens progressively in the fifth, 
 fourth, etc., while in the dissonances we find least of it. Consonance 
 and dissonance appertain in this original and limited sense only to 
 every two tones. " Only as thus understood, as the relation of two tones 
 to one another, is consonance the basal phenomenon of all music" 
 (71, 329). "It must always be borne in mind that what I call fusion 
 can only then be perceptible as such when the fusing tones are distinguished 
 from one another; just as we cannot perceive similarities without keeping 
 the similars separate. But if this is done, if the three tones of a trichord 
 are distinguished from one another, I at least can form a judgment on 
 their fusion only by pair-wise comparison, but I cannot besides discover 
 a fusion that attaches to the whole, to the triad as such" (71, 330). 
 
 It must be evident that thus far at least the empirical teachings
 
 xvn] COMMON CHORDS OR CONCORDANCE 143 
 
 of harmony gathered together in the previous pages and the results 
 that have emerged from them confirm this general attitude of Stumpf's 
 towards intervals quite unambiguously. Every pair of tones is in 
 harmony a distinct individual, as it were; it in no way ceases to be 
 itself or changes into another, owing to the simultaneous presence of 
 other tones. As that individual, it carries its own degree of 'fusion' 
 unchangingly about with it, although, and this must be emphasised, 
 the effect produced by that fusion at any moment is to some extent 
 modifiable by a number of circumstances other than the fusion itself. 
 Moreover the appreciation of all harmonic effects, even of such elementary 
 ones as we have as yet been able to study, presupposes always in every 
 musical ear some sort of ability to distinguish every pair of voices from 
 every other. 
 
 What kind of distinction is implied is left unsaid. Doubtless it may 
 vary greatly in degree of clearness. Low grades of distinction, as musical 
 analysis of finer order would rate them, are apparently quite good 
 enough; for every beginner is supposed to be able to appreciate readily 
 enough what is taught. The ease and certainty of analysis that is 
 habitual in the most finely endowed musical minds is by no means 
 essential. The beginner is not required to be able to name every ordinary 
 chord as soon as heard, or to sing its components, or even to hear them 
 by mental analysis singly, or to separate in turn each pair from the 
 others in his mind's ear. It is enough if he can hear and attend well 
 enough to get the chief effects that are produced by any pair of voices 
 amongst others : e.g. the bad effect of consecutives, exposed intervals, 
 etc. That, the teachings of harmony show us implicitly, is already 
 hearing the tones of chords pair by pair 1 . 
 
 Another point stressed by Stumpf is that "consonance is not changed 
 either by the addition of a third or fourth tone. What is changed is 
 the musical meaning of the tones and their pleasantness. But the 
 unitariness of the octave, the duality of the seventh survives in any 
 and every arrangement with other tones" (71, 328). That, again, is 
 true, but only in the sense of the preceding paragraph. The original 
 essence or being of the consonance or dissonance is not altered, but the 
 effect of it or its suitability or its functions as a unit of musical structure 
 are certainly changed. The terms used by Stumpf the musical meaning 
 of the tones and their pleasantness do not specify what these functions 
 
 1 Cf. 72, 51-57, which still fails to bridge the gulf between the usual static analysis 
 of tone-masses and the fluid analysis of music.
 
 144 COMMON CHORDS OR CONCORDANCE [OH. 
 
 are. They imply, however, that the degree of fusion is not their basis 
 or source. 
 
 The complicated psychical processes that bring certain modes of apprehension 
 to bear upon sensations that have been changed either only subliminally or not at 
 all (often even in a contrary sense) must not be confused with the simple facts of 
 sense perception by which the basal phenomenon of all, even of non-harmonic, 
 music is given. That one and the same unmodified pair of tones should now fuse 
 more and now less according as we apprehend it as c el? or as c dj is out of the 
 question, because fusion is a function of the two sensations or of their physiological 
 bases and can change only with these same (71, 328). 
 
 In view of the needs and practices of music Stumpf's attitude towards 
 his notion of fusion is readily intelligible. A generalised notion of fusion 
 in the sense of degree of unitariness, applicable to any tonal mass of 
 however many components, would fail to solve the problems of musical 
 science. Stumpf does well to look about for some new fundamental 
 notion that will meet the situation. Nevertheless it remains true that 
 the notion of unitariness is logically quite as applicable to any number 
 of simultaneous tones as to two. A triad cannot but approximate more 
 or less to the unitariness of a single tone, even if we add the proviso : 
 whether its component tones are distinguished or not. If that approxi- 
 mation and that proviso pass for two tones, they must be equally valid 
 for three or more. Stumpf's attempt to dam up the logical vitality of 
 the concept of fusion is certainly not the method that will lead us 
 quickly forwards. The procedure makes a semblance of success only 
 so long as the waters fail to overflow. 
 
 But let us notice the alternative foundations offered : 
 
 Our music rests without doubt upon the trichord in its two forms major and 
 minor. The question then is: what is the objective justification, the reasonable 
 principle of structure, of trichords? This question is usually either not asked at 
 all (as in the most of the text-books of harmony) or it is absolved by a reference to 
 the series of partials. In this series 4:5:6 are indeed found, and further on the 
 minor trichord 10 : 12 : 15 as well; but there are in it many other trichords besides, 
 that are not honoured in such a way by music, although they partly have even 
 smaller ratios than the minor, such as 7:9: 11. What then gives these two chords 
 their dominating position, and why must just three tones be bound together gene- 
 rally, if more than one are to be combined at all? 
 
 The fundamental principle may be formulated thus : Let the greatest number 
 of tones within the octave be taken that are severally consonant with one another, 
 and so that we pass in the tonal motion from below upwards and amongst the 
 consonances from the stronger to the weaker degrees of consonance. 
 
 Starting from any tone we get according to this principle first its upper fifth, 
 so, from c, g, and then only either et> or e is further possible, if we neglect for 
 the present the 'sevens' [5 : 7 and such like]. Thus with the upper finish of the
 
 xvii] COMMON CHORDS OR CONCORDANCE 145 
 
 octave there result the two chords cettyc 1 and cegc 1 . In them all higher grades of 
 fusion are represented. But as it is at once apparent that c 1 has again an octave 
 above itself and within this new octave-space the same process repeats itself, there- 
 fore we do not reckon c 1 further as a part of the structure won from c, but as funda- 
 mental tone of the analogous one an octave higher. Thus we reach the trichord, 
 in its two forms simultaneously (71, ssit.). 
 
 Stumpf then proceeds to show in a very summary way how the 
 usual chordal combinations of modern music might be developed. 
 Finally he gathers the results together in special concepts. 
 
 As a chord we designate a group of simultaneous tones... that can be reduced 
 in the way indicated to chief or accessory triads of a certain fundamental tone. 
 Tone-groups, therefore, with dissonant intervals are called chords in this sense,, 
 but not all and sundry, only those that can be obtained from triads by certain, 
 operations (71, 337). 
 
 Chords, therefore, fall into two classes. Concords (as in our usual 
 sense of the word) must contain a fifth or its inversion a fourth, and a 
 third or a sixth. Discords are all other chords in the sense just expounded. 
 Concordance and discordance are the corresponding abstract terms. 
 
 Consonance and dissonance are thus presupposed by the notions 
 of concordance and discordance. But the latter notions differ from the 
 former, which apply only to pairs of tones and to tone groups only in 
 so far as they consist of pairs of tones. Concordance applies primarily 
 only to groups of three or more tones and can be transferred to a tone 
 pair only if and in so far as it is apprehended as a part of a concord, 
 i.e. of a triad (71, 340). Thus one and the same tone pair may be at 
 one moment concordant, at another discordant according to the setting 
 in which it is apprehended. It is the setting that makes the difference. 
 So concordance and discordance only appear with at least three tones. 
 And "consonance is a matter of direct sensory perception, whereas 
 concordance is a matter of apprehension and relational thinking " (71, 341). 
 
 Stumpf points out finally that these expressions are not by any 
 means new to musical theory. " But since Franco [the words concordare, 
 discordare have been used] perhaps from the feeling that it is no longer 
 a matter of merely 'sounding together or sounding apart,' but also of 
 'fitting together and not fitting together'" (71, 350). 
 
 Now, however valuable this exposition of the notion of concordance 
 may be in so far as it gives an account of the character peculiar to 
 groups of at least three tones or to intervals as parts of these (compare 
 the notion of 'pattern' expounded in chapter x above), certain points 
 call for immediate remark. 
 
 w. r. H. 10
 
 H6 COMMON CHORDS OR CONCORDANCE [CH. 
 
 (1) Upon what real ground of tonal functions does the alleged 
 constitutive principle of chords rest? None has been given or even 
 indicated. A logical ground alone is evident. That, of course, is in 
 itself a very important matter, but it is quite powerless to make between 
 tone pairs and triads the real separation that Stumpf claims. It could 
 at most make a merely logical division, such as would divide the discus- 
 sion or study of tone pairs from that of triads. It could not justify 
 the rule that consonance applies only to tone pairs, concordance only 
 to triads or larger groups of tones. Nor could it do anything to constitute 
 the relational thinking that is claimed as the essence of concordance. 
 It remains as great a mystery as ever how triads with their three tone 
 pairs come to form the basis of modern art. 
 
 (2) No doubt Stumpf is firmly convinced that consonance is always 
 a function of two tones at a time. And he may well be right in this. 
 But even then he is so only through 'knowledge by acquaintance,' 
 not through* 'knowledge by description.' In other words he feels it or 
 knows it by experience, but he does not know it logically or scientifically. 
 It has not been proved by him. On the contrary the principle upon 
 which concordance is founded would lead us to expect that a chord is 
 only a group of fusional pairs or their derivatives. Then there would 
 be no real division between groups of two, and groups of three or more 
 tones. And there is also no clear reason why we should not turn our 
 relational thought upon a succession of tone pairs as well as upon a 
 sequence of chords or look upon concordant triads as parts of discord- 
 ances in four or more voices. There seems to be no such radical distinction 
 between music of two parts and music of more than two parts as Stumpf 's 
 distinction between consonance and concordance would lead us to sup- 
 pose. 
 
 (3) If two tones necessarily make some approximation to the unity 
 of a single tone of whatever degree, we have still as far at least as 
 Stumpf's science can show every reason to expect that any group of 
 tones should do the same. Or rather we should expect that every triad 
 should be rather more dissonant than otherwise. For it would certainly 
 not tempt us to take it for a unity, whether we distinguished its com- 
 ponent tones or not. 
 
 (4) Here we come upon an important point. Stumpf's dissonance 
 is more or less a negative idea 1 , like Helmholtz's consonance in 
 
 1 As it is also in many other writers, amongst the ancient Greeks for example, and 
 in Gevaert himself (cf. above, p. 108). But not all the Greek writers neglected the positive 
 aspect of dissonance (v. p. 154, below).
 
 xvn] COMMON CHORDS OR CONCORDANCE 147 
 
 simultaneous intervals. It is merely a minimal degree of approximation 
 to unity, of unitariness (with or without distinction of the component 
 tones or pitches). Is that enough! Affirmation would imply that two 
 minimally unifying tones are as such unpleasant. But why so? There 
 is no obvious reason. On the other hand, if while highly unifying 
 pairs give approximation to balance, minimally unifying pairs give, 
 not merely a non-unity, but a positive unbalance or irregular confusion, 
 we should be able to bring that chaos as a positive ground of unpleasant- 
 ness into connexion with similar grounds in other spheres, e.g. pictorial 
 art, logical thought, feeling, etc. 
 
 But that is not the end of the subject. Evidence has been brought 
 above to show that there are grades of fusion that must be called 
 neutral neither distinct consonance nor distinct dissonance. Here we 
 come upon an aspect that does not seem to be subsumable under the 
 fundamental notion of fusion as approximate unitariness. Nor can we 
 well conceive of an indifference-point between balance as approximation 
 to the unity and symmetry of a single tone and unbalance or chaos. 
 How far away from balance should we have to fix this point? But 
 one might say : consider the middle point to be balance and suppose a 
 departure from it in two directions, one towards loss of balance in unity, 
 the other towards loss of balance in conflict. At both extremes we tend to 
 lose sight of the component tones. In consonance they run too much 
 into one another, in dissonance they obscure and confuse one another 
 too much. 
 
 Such a view would not quite coincide with the useful grading of 
 fusion from a maximum to a minimum on the basis of unitariness. 
 But that would be no insuperable barrier. We might still conserve this 
 grading as a partial aspect of the problem and at the same time prefer 
 the other as more adequate to the sensory stuff. Loss of distinction 
 in unity, balance of distinction, and loss of distinction in confusion can 
 certainly be logically represented as a series from a maximum to a 
 minimum as a decrease from approximation to the balance of a single 
 tone, and therefore valid for scientific purposes. But for musical 
 purposes the other notion which centres upon the point of balance of 
 distinction seems by far the more important. 
 
 For it simply lays upon our hands the solution of the problem of 
 the great and fundamental importance of the thirds and sixths in all 
 music and of the triad in modern music. The thirds and sixths are the 
 intervals of greatest balance of distinction of tones. Two or more 
 
 102
 
 148 COMMON CHORDS OR CONCORDANCE [cu. 
 
 thirds or sixths after one another, therefore, also afford as easy distinction 
 as one. Hence we pass immediately to the interpretation of this 
 distinction as melodic distinction. And so a series of thirds or sixths 
 is most favourable to melodic continuity, as the whole system of facts 
 gathered together in the previous chapters have shown us to be the 
 case. 
 
 The importance of the triad for music therefore lies in the two 
 thirds it contains. And, of course, an alternative is created by the two 
 possible positions of the major and minor thirds in each common chord. 
 Thus we get the major chord for the one and the minor chord for the 
 other. Two other possibilities exist, namely the triads containing two 
 minor thirds or two major thirds. These, however, each contain another 
 important interval. The tritone of the one is a distinct dissonance. 
 The (augmented fifth or) minor sixth of the other is not ordinarily a 
 dissonance, but a neutral interval, a maximal balance of distinction. 
 But in the triad it always acts as a dissonance. Many reasons might 
 be suggested for this. We need not attempt to find the most probable 
 one at this point. Having been carried thus far by both fact and logic 
 we may claim to recognise as fact that the fundamental triads both 
 contain a fifth between the outer tones of their two thirds. Upon 
 these two triads all harmony is said to revolve. But the other two 
 triads also are in common use. The distinctive feature of the common 
 chords is due to the fifth they contain. This high grade consonance 
 gives the whole a special unitariness or stability; but this aspect of 
 things we shall leave for special treatment in a later chapter. 
 
 The same principle that explains the essence of the common triads 
 will account also for the discords that form so important a part of 
 modern music. If two neutral intervals may be combined to form a 
 triad, it follows as a matter of course that three or more may be combined 
 to form greater chords. These will always be discords. For the repetition 
 at the octave of any one of the components of a common triad gives, as 
 we have seen, merely an extension of the characteristic whole or pattern 
 formed by the three essential tones. Thus we obtain a set of chords 
 in which all the possibilities of combination of thirds may be exhausted. 
 Many other ranges of possible combinations may be taken into view, 
 if all the possible inversions of groups of three thirds are examined 1 . 
 
 The musical utility of any of these chords will, of course, depend, not 
 
 1 The deduction of these paragraphs is not meant to imply that only chords derived 
 from columns of thirds are to be countenanced. Of this we shall see more as we proceed.
 
 xvn] COMMON CHORDS OR CONCORDANCE 149 
 
 so much upon the neutral nature of the thirds that make up what has 
 been held to be their original position, as upon the kind of intervals 
 that are actually formed between each pair of voices that appear in 
 the chord. It would be a mistake to take any interval or class of intervals 
 as the primary ground of a chord to the disadvantage or depreciation 
 of any other. All the intervals that occur in a chord are of equal 
 importance, except the octave (for the reason we have given). When 
 a seventh or a second occurs, it has the fusional status of a second or 
 of a seventh, and by no means that of the third that may ensue upon 
 its inversion or in relation to some other tone of the chord than the 
 bass of the interval in question. 
 
 The view we thus obtain of the part played by thirds in the establish- 
 ment of chords falls into line with the empirical principle that was 
 extracted by Rameau from the aesthetic work of music and that has 
 been used and defended repeatedly since. This is "the theory of the 
 generation of chords by adding thirds together" (60, 81). It has indeed 
 never been proved in any sense of the term (cf. above, end of chapter x). 
 But it has always made a strong claim to recognition merely by the 
 force naturally inherent in it, apart from all theory, as a generalised 
 expression of empirical practice 1 . And on this ground it must be held 
 to be far more worthy than all the attempts to found a system of chords 
 upon the resonance of the sonorous body or upon the series of partial 
 tones. 
 
 Much time and energy has been wasted upon the problem of the 
 fundamental chord or chords from which all the others are derived, 
 upon the systematisation of chords for the purpose of rinding their 
 origin, and such like questions. Certainly it was extremely important 
 for musical study to discover the connexions between chords that we 
 know as inversion. That achievement is a piece of direct and unshakable 
 description, which musical theory has to explain in some way or other. 
 And it was equally valuable to work out the differences between systems 
 of intervals and chords, that are summarised in the distinction of major 
 and minor modes. But it is as absurd to put one chord down as the 
 origin of another, as it would be to consider a single interval, or a single 
 tone as the one and only progenitor of all. Besides any chord whatever 
 
 1 Well expressed by M. H. Glyn (18, *&): "The third has always been beloved by the 
 natural ear. We have to deal here with a fact of far greater importance to music than any 
 in the science of acoustics, and if consonance to music means the third, and only in a 
 limited degree the fifth and the fourth, while to science it means the fifth and the fourth 
 and after that the third, it is clear that two points of view are being named by the same* 
 name which are by nature different and should be recognised as such."
 
 150 COMMON CHORDS OR CONCORDANCE [CH. xvn 
 
 can be reduced to a column of thirds by suitable transposition of its 
 tones through octaves. That follows from the fact that a continuous 
 column of thirds (major or minor) soon yields all the tones of the 
 chromatic scale. Thus, c, e, g, b, d 1 ,/ 1 , a 1 , c 2 = the diatonic scale; the 
 notes of the chromatic scale can be got by suitable substitution of 
 minor for major thirds and conversely. 
 
 A musical experimentalist is free to form any interval he can upon 
 his instrument. If it is good and useful he will introduce it wherever it 
 will produce or enhance a desired effect. But he must above all make 
 it possible for the listener to hear it properly; and that enforces the 
 limitation of the number of chords and of their positions in the tonal 
 range. The whole history of music is an attempt to find a system of 
 tones which will yield the greatest variety of chords and the greatest 
 number of relations between them that in turn will most facilitate the 
 apprehension of the tones played and make possible the greatest scope 
 and freedom of aesthetic effects. 
 
 It is easier and more natural to bear in mind the actual development 
 of music from the earliest times and to see how the science of musical 
 sounds develops towards more and more complete explanation of its 
 course than it is to try to derive music from an unintelligible genealogy 
 of its final products. The musical mind of the world did not begin 
 under the inspiration of a subconscious appreciation of all musical 
 effects. It began with a very limited sense or feeling for these things. 
 But enjoying what it already had, it strove to make that little grow to 
 greater ends. And as it laboured, the effects were formed experimentally 
 and the ear seized them. The growth of the art gradually revealed 
 more and more subtle aspects and wider and wider connexions or 
 systems of effects. Two of the greatest of these are polyphony and 
 tonality. These were not given; they had to be discovered. Art is as 
 much a process of discovery as science is. Both are experimental and 
 systematic. But while art is content to be empirical, science is restless 
 till it has grasped the whole system of inner bonds that rule its objects 
 and has described them fully and thoroughly.
 
 CHAPTER XVIII 
 
 MELODIC MOTION IN RELATION TO DEGREES OF CONSONANCE 
 
 THE analysis of the degrees of fusion we have just given may be extended 
 in a way that seems to be of importance. 
 
 We have distinguished three chief grades : loss of distinction in 
 unitariness, balance of distinction, and loss of distinction in confusion. 
 And we have noticed that balance of distinction must make for ease 
 and continuity of melody, when two or more melodies run side by side. 
 The question that now promises to further our insight into the structure 
 of music is : what effect has loss of distinction upon melodic continuity, 
 upon the ease and distinctiveness with which two or more melodies 
 will run side by side? 
 
 High grade consonance we have already learnt to look upon as 
 balance, approximation to the unity of a single tone. Its component 
 tones are wrought together more than usual; they cling together and 
 do not offer to pass with as much ease into two other tones as they 
 were approached from those that preceded them. They constitute, 
 therefore, a point of relative rest and tend to bring the voices to a stop. 
 
 The same effect is produced under certain circumstances by the 
 unison, as we have seen above (p. 130). A unison is, of course, from an 
 absolute acoustical point of view a single tone, certainly not an interval. 
 And a single tone is not as such in music arrestive in function. Unison 
 has a significant meaning only in so far as two melodic streams are 
 felt to meet and to be identical in a certain sound. Being usually two, 
 they then 'sound one.' When such a unison is presented without 
 correction by the various circumstances brought to bear upon high 
 grade consonances to make them mobile, it produces the same undesirable 
 effect as they do. Two voices are caught up into one and an effect of 
 arrest and confusion is produced by the loss of distinction in unity. It 
 is only in this way that the functional similarity of unison and high 
 grade consonances can be justified. 
 
 This function of stability and arrest 1 peculiar to the consonances 
 
 of the octave and fifth has long been recognised and shows itself in 
 
 various ways. In Greek and early Western music the octave (or unison) 
 
 was the usual close of a piece. The root position of a triad is a more 
 
 1 Or 'repose': cf. D. F. Tovey, 74 passim.
 
 152 MELODIC MOTION IN RELATION [CH. 
 
 stable form than either of its inversions; the fifth in it spans the two 
 thirds or gives at least (in the alternative arrangement) a fifth with the 
 bass. In the first inversion there is either no fifth at all or in the alternative 
 arrangement only between the two upper voices. (Conjunction with the 
 bass we have already seen to be generally more powerful than conjunc- 
 tion with the soprano.) In the second inversion we find the peculiar 
 feature of a fourth from the bass. As a consonant interval that will 
 produce some effect of rest corresponding to its grade, which approxi- 
 mates towards the neutral range, though still above it. Of these three 
 forms of the common chord the first is, as the older theorists said, 
 ' most apt to conclude ' ; it produces the greatest arrestive effect upon 
 the flow of voices. When this effect is heightened by special means by 
 the use of the chief transitions of tonality, dominant or subdominant 
 to tonic, and by suitable rhythmical exposure, etc., we get the various 
 cadences, whose sole function is to produce partial or complete arrest. 
 Other contributory, and therefore functionally similar, conditions are 
 a gradual reduction of speed, a greater sonorousness and steadiness in 
 tone production, the repetition of the final chord, and so on. 
 
 The counterpart of the high grade consonances is formed by the 
 high grade dissonances. These create a loss of distinction in confusion, 
 which must also have a disturbing effect upon the ease and continuity 
 of melody. The treatment of dissonances in music is a natural con- 
 sequence of this. If dissonances are to be introduced, special means 
 must be employed to overcome the loss of distinction as far as possible. 
 Devices, such as suspension and preparation, were early invented and 
 rigorously prescribed. Although they are not now considered to be 
 indispensable, they have by no means been superseded as superfluous. 
 They serve to fix in advance in the hearer's mind the most difficult 
 part of the group of tones about to be presented, whereupon the others 
 are introduced by common and easy melodic procedure. In this way 
 the listener is enabled to follow the movement of all the melodies equally 
 well, and so is shielded from the confusion that might otherwise arise. 
 Besides this subtractive method of approaching dissonances, there is 
 of course the method of contrary motion by which the tones forming 
 a dissonance with one another may be approached from opposite sides. 
 The listener is thus guided carefully through the moment of confusion. 
 
 The musical mind of to-day has grown so familiar with all the 
 melodic combinations our harmonic procedure has reduced to distinct 
 types, that there are many who almost suggest that in the course of time,
 
 xvm] TO DEGREES OF CONSONANCE 153 
 
 as music progresses, what was previously discord comes to be reckoned 
 as concord. Stumpf said: "Effect upon feeling is specially liable to 
 change, even within our system, in that the unpleasantness of discords 
 weakens and through the introduction of new and ever bolder discordant 
 structures the old ones take on the feeling effects of concords; so that, 
 as v. Hornbostel remarked, progressions to these old discords can act 
 soothingly like a resolution into concords" (71, 341 f.). Stumpf, however, 
 does not think that such changes of feeling will ever break down the 
 difference between concord and discord. We may well agree with him; 
 for a definite reason can be given that seems to be of substantial weight. 
 
 However familiar we may become with the patterns of discords, 
 that will surely never in any way alter the graded differences there are 
 between consonances, neutral fusions, and dissonances in respect of 
 balance of tonal distinction. Both the current theory of the derivation 
 of all intervals from the series of partials and Stumpfs theory of fusion 
 grade the intervals in a series of indefinitely decreasing consonance. 
 The one end of Stumpfs series, as we have seen, is characterised by 
 apparent unitariness of sound, the other by closer and closer approxima- 
 tion to mere clear-cut apprehension of two-ness. The theory of partials 
 suggests that the nearest and loudest and perhaps most frequent 
 partials yield the consonances that are distinguished earliest in the 
 history of music, and that as music advances our familiarity with 
 partials extends farther along the series, so that we reckon as consonances 
 always as much as we have thus made our own. The history of music 
 seems to provide a parallel to this in the early use of the octave, the 
 subsequent 'organising' in fifths and fourths, and the later classification 
 of thirds and sixths, or even sevenths as consonances. But, on our 
 interpretation, all this line of speculation is completely cut out. The 
 series of fusions has its neutral point or its region of natural ease and 
 familiarity, as it were in the middle, in the thirds and sixths. From 
 this point the difficulty of manipulating the intervals in polyphony 
 or in harmonic music increases in the two opposite directions towards 
 the consonances and towards the dissonances. Familiarity may give us 
 greater facility in dealing with these naturally recalcitrant intervals; 
 it may even induce us to dispense with certain aids to apprehension 
 that we once found necessary or desirable. But it cannot alter the 
 natural differences between the various grades. 
 
 Once we have found the true system of functions of intervals, the 
 false motive suggested by the apparent course of history entirely loses 
 its value. The historical order of approach is quite irrelevant and can
 
 154 MELODIC MOTION IN RELATION [CH. 
 
 be readily explained otherwise. It was the prevalence of monophony 
 that led to the adoption of the octave first of all intervals; monophony 
 does not essentially change in becoming homophony. And homophonies 
 in fifths and fourths are the next inevitable attempts at continuous 
 development. The great consonances call early attention to themselves. 
 But it is only in polyphony that the polyphonic functions of these and 
 all the other intervals can be discovered. And it is these functions of 
 intervals whereby they either yield a simultaneity of easy flowing 
 melodies or disturb one another in this respect that determine the 
 final classification of intervals. 
 
 As for the plea that the Greeks had not yet recognised the consonance 
 of the thirds and sixths, the evidence seems to indicate merely that 
 they did not reckon these intervals among the distinct consonances. 
 Neither do we really. They do not show a notable degree of approxima- 
 tion to unity ("so that the resulting sound is one like and similar to a 
 single one," Nicomachus 71,329; 66,54). That still leaves room for 
 two other classes, one in which the two sounds, far from being one- 
 like, are rather specially two-like, shall we say? or diaiphomc 1 , 
 discordant, contraposed ("when the sound of the two is as it were 
 rent asunder and without true blending," Nicomachus 66, 54); and 
 another middle one in which the two sounds are just two, neither 
 friends nor enemies, but just comrades. 
 
 Knowledge by acquaintance may change then ; and so may knowledge 
 by theory, and practice, and familiarity and all such adjuncts of feeling 
 or sensory experience ; but sensory feeling itself does not seem to change. 
 The conformations of sense retain their characters unaltered. Sense 
 is a stuff that the growing mind of man may learn to mould as he can, 
 but ever in obedience to the laws inherent in it. It is as much an objective 
 world that we must learn to know and to use as is the world of 
 nature. 
 
 1 Cf. 14, %: "La sensation auditive produite par les consonnances et les dissonances 
 est analysee d'une maniere uniforme par tous les ecrivains: "dans la consonnance les 
 deux sons se melangent au point de s' absorber mutuellement, de telle maniere que 1'oreille 
 ne re9oive qu'une impression unique, douce et suave." Elien le platonicien compare la 
 consonnance a "du vin mele de miel, ou aucune des deux substances ne predomine, et 
 ayant le gout d'un breuvage particulier, qui n'est ni du miel ni du vin. Dans la dissonance, 
 au contraire, le melange ne s'opere pas; les sons se repoussent, pour ainsi dire, 1'un 1'autre, 
 et 1'impression totale est dure et penible." Of course the idea in "ni du miel ni du vin" 
 must not be pushed to the extreme of positing a new resultant third tone (cf. 16, issf.; 
 66, 52). "Aussi les definitions antiques de la symphonic et de la diaphonie sont-elles en 
 grande partie sanctionees par 1'oreille moderne" (16, iss).
 
 xvm] TO DEGREES OF CONSONANCE 155 
 
 But to pass on. Dissonance must differ from consonance not only 
 in the way described, but in another subtle manner. If consonance 
 creates the effect of a pause or rest by presenting us with an approxima- 
 tion to unity, and if we accept the suggestion to rest, or if it does not 
 conflict with the effect of the other tendencies of sound at the moment, 
 but forms a consistent whole with them, then we shall be somewhat 
 careless of distinguishing differences within the unitariness. We are then 
 either wholly or relatively at rest and we do not need to be scrupulous 
 in distinguishing. We have no need for such finished distinctions, for 
 we do not crave to move forwards. If, on the other hand, we want 
 the music to move forwards through a consonance, in spite of the 
 tendency of consonance to create repose, we must be careful not to 
 strengthen the reposeful effect by any means of the same tendency. 
 Hence flow the rules against consecutive octaves and fifths, and the 
 exposure of these intervals. 
 
 In dissonances, on the contrary, a point of unrest is created. There 
 is neither rest nor even flow, but a disruptive effect, and disagreement 
 between the component tones. They get in each other's way and produce 
 mutual confusion and disturbance. Even if we have been led skilfully 
 into the dissonance, we nevertheless are impelled forwards. We look 
 for another phase of progress in which disturbance shall cease. But 
 not any consonance (or dissonance) may follow; only one which we 
 can easily and melodically reach from the present dissonance. Thus 
 the need for resolution of dissonances arises. When several voices run 
 concurrently, and there is consequently more to follow and more danger 
 of losing the thread of sequence, the need for resolution is all the greater, 
 and must be the more carefully controlled. Thus it appears that dis- 
 sonance, far from being a barrier and a hindrance to good music, acts 
 clearly as a stimulant upon melodic activity, urging it forwards and 
 increasing expectation of progress. 
 
 Much might be said in favour of the adoption of the common Greek 
 terms 'symphony' and 'diaphony' with the addition of the rarer 
 term 'paraphony.' In symphony the tones of an interval tend to become 
 indistinguishable through too much unitariness or fusion; in diaphony 
 they sound through or against one another, disturbing and confusing 
 one another; in paraphony there is balance, so that melodies formed 
 of such intervals will flow evenly side by side, the one not inhibiting 
 the apprehension of the other. Paraphony, it should be noted, does 
 not imply that the tones of such an interval are on the whole more
 
 156 MELODIC MOTION IN RELATION [CH. 
 
 easily apprehended as mere duality than the tones of a diaphony. 
 In the latter the duality is indirectly emphasised by the harsh confusion. 
 Paraphony implies merely a medium grade of obviousness of plurality 
 through lack of approximation to the unity of a single tone (in the 
 sense of Stumpf's fusion), but at the same time a maximum grade of 
 distinguishability for those musical purposes which require distinction 
 of pitches and apprehension of melodic flow in more voices than one. 
 For these purposes we require a perfectly clear-cut untroubled dis- 
 tinguishability of the component tones of an interval. That is plainly 
 wanting at the consonantal end of the fusional series. It is merely 
 presumed to be present at the dissonantal end, because at that end there 
 is the more obvious indication of two-ness of tone; the tones jar upon 
 one another harshly and there may be very obvious beating between 
 them ; even the unmusical mind reads there signs confidently as two-ness, 
 while the musical ear finds in the dissonances more obvious lack of 
 fusion or presence of ruption than in the thirds and sixths. But it would 
 be right to claim that in the small dissonances it is more difficult to 
 pick out the pitches of the component tones than it is in the thirds and 
 sixths, and that in the larger dissonances, although the pitches stand 
 well apart, yet the two tones do 'oppose' one another or jangle with 
 one another 1 , besides being very much two-like apart from that. Thus 
 the validity of the fusional series, as we find it in Stumpf and others, 
 would be called in question. In music an abstraction of melodic function 
 from apparent one-ness and jarring two-ness has been carried through, 
 which points to the neutral thirds and sixths as the region of pure 
 two-ness of tone in interval. The clear distinction of melodies must 
 therefore be but a seriation of this two-ness function, not a new kind 
 of (perpendicular) distinction of tones supervening upon their melodic 
 combination. It is a task for experimental work to find and to describe 
 exactly the analytic attitude that will confirm this conclusion from the 
 functions of intervals in polyphonic music. 
 
 Our new results may be summarised very briefly with the aid of 
 these terms. The movement in music is melodic. For the proper flow 
 of simultaneous melodies intervals must either be themselves actually 
 paraphonic or they must be used paraphonically. 
 
 Note. Of the Greek terms relating to harmony, symphony and 
 diaphony are by far the most familiar and have gone over simply 
 
 1 In particular the lower boundary of the higher tone falls near to, and so defiles, 
 the symmetrical outline of the lower tone in its most important point its pitch.
 
 xvm] TO DEGREES OF CONSONANCE 157 
 
 into the terms consonance and dissonance. I have quoted characteristic 
 examples of their definitions above (p. 154). The term paraphony was 
 used by several later writers, Thrasyllus, Bacchius and Gaudentius 
 (66, 48f., 67 ff.; 16, 139). While the relevant passage in Bacchius is almost 
 certainly confused, Thrasyllus attached the term to the fifth and fourth 
 in distinction from the octave, a subdivision which does not seem to 
 be of any particular interest, as far as we are concerned nowadays. 
 But the relative passage in Gaudentius is of the greatest importance. 
 It has apparently been a source of mystification for most interpreters. 
 Stumpf speaks of it as "this otherwise (i.e. than by his explanation) 
 quite incomprehensible passage" (66, 72) 1 . 
 
 Since my conclusions call so clearly for the use of a term like ' para- 
 phony,' it becomes a matter of much interest to consider whether 
 Gaudentius may have been led to his somewhat similar application of 
 the same term from a similar train of thought. Apart from the mere 
 existence of the term, its verbal meaning, and the conceptual setting 
 of the notion between symphony and diaphony, I can find with the 
 help of the chief authorities (Gevaert and Stumpf) little or nothing to 
 show what the idea of Gaudentius really was. Stumpf suggests merely 
 that Gaudentius meant the major third and the tritone to be taken 
 simply as "consonances of lower grade, as transition to the dissonances" 
 (66, TO). 
 
 Stumpf's translation of the passage in Gaudentius is as follows : 
 
 Symphonic are those in which, when they are simultaneously struck or blown 
 on the flute, the melos of the lower in relation to the higher or conversely is always 
 the same, or (in which) as it were a fusion in the performance of two tones occurs 
 and a kind of unity results. Diaphonic are those in which when they are simul- 
 taneously struck or blown, nothing of the melos of the lower in relation to the 
 higher or conversely appears to be the same or which show no sort cf fusion in relation 
 to one another. Paraphonic are those that, standing in the middle between the 
 symphonic and the diaphonic, yet appear symphonic when played; which seems 
 to be the case in the tritone (/-&) and the ditone (g-b), (66, 69). 
 
 Elsewhere (66, 66) Stumpf says that melos is "perhaps best translated 
 as 'the melodic element of tone' or the tonal element of melody.'* 
 
 No doubt the matter might be discussed at great length. But if 
 we leave the first clauses of Gaudentius's definitions as obscure, and 
 (or) interpret them by the following clauses, then we must look upon 
 Gaudentius's use of the term 'paraphony' as well founded. Symphony 
 is the greatest fusion (when the melos of the two tones tends to sameness), 
 
 1 Cf. 14, 99: "Mais il importe do remarquer quo cotte doctrine cst isolee dans la litt^ra- 
 ture iim-ii-.il.- des anciens."
 
 158 MELODIC MOTION IN RELATION [CH. 
 
 diaphony is the least fusion (when the melos of the two tends to extreme 
 difference), paraphony is the middle between these, which appears 
 as symphony in performance. It is probably this last clause that has 
 compelled writers to think Gaudentius meant this middle class to be 
 a lower grade of consonance, not a really middle neutral class, and so 
 to miss perhaps the main point of Gaudentius's distinction. That main 
 point is the placing of the neutral relation (para) in the middle of the 
 whole series and the attaching of it in particular to the major thicd. 
 It is much more significant than any distinction between the octave 
 and the fifth with the fourth. Music naturally gives the octave a special 
 place, because it is its unit of division, whereby repetitions in octaves 
 become practically mere repetitions or identities. But the grading of 
 octave, fifth, fourth had been established long before Gaudentius (by 
 Aristoxenus, 66, 38). The former did not add the major third to them 
 as a lower grade of consonance, but as a member of another class lying 
 between the consonances and the dissonances. 
 
 It may seem perverse to stress the point; but it is really an important 
 one. Music makes a distinction of opposition between consonance and 
 dissonance, but it has failed to recognise that distinction theoretically 
 in so far as it ranks the thirds and sixths as imperfect consonances, not 
 as neither consonances nor dissonances but neutral 'sonances' or 
 paraphonies. Gaudentius makes this division, adding the note that 
 paraphonies sound symphonic in (instrumental) performance (eV ry 
 /cpov<rei). 
 
 In a sense, our present musical classification may be said to be 
 just the obverse of the early Greek one, primitive and limited as that 
 has been usually thought to be by modern writers. The older Greeks 
 made one dividing line below the fourth, calling all the rest diaphonies. 
 We draw our great line below the thirds and sixths, calling all except 
 the dissonances consonances. Let us combine the two and we get 
 in principle the divisions of Gaudentius. And if we like, we can add 
 with him that eV rf} Kpov<rei in a mere interval of two tones struck 
 together as distinct from two melodies (as we inay say) the paraphonies 
 seem symphonic. In other words when it is not a matter of paraphony 
 generally, but just of the general character of a chord as a whole, we 
 reckon the specific paraphonies (thirds and sixths) to the symphonies. 
 Or, we might say : we take them not as apparently one-like like the 
 high grade consonances, but as agreeing with one another and therefore 
 as consonant, because they do not obviously jar upon one another 
 as do the distinct dissonances.
 
 xvm] TO DEGREES OF CONSONANCE 159 
 
 Of course it would be improper to read into Gaudentius all that we 
 can now put into the skeleton of the distinctions to fill them out. But 
 he should have the benefit of any doubt there may be. His conceptual 
 scheme is full enough, but there is not enough detail. 
 
 All modern theorists have treated Gaudentius's distinction .as if it 
 merely amounted to an extension of the grades within the class of 
 consonances by a further step downwards. That surely does violence 
 to his words and to his term (paraphony). If this modern view is wrong, 
 then with it must go the attempt to see an evolution of the notion of 
 consonance downwards from the octave, to include first the fifth and 
 fourth, then the thirds and sixths, now the natural seventh and tritone, 
 and to-morrow all the dissonances themselves (cf. 22, us). Impossible ! 
 That were no evolution, but a debasement. Evolution unless it be 
 the degeneration of the parasite that casts off its sense-organs means 
 progress, an increase in the complexities or in the differences distin- 
 guished, not the swamping of all differences in one class 1 . All differences 
 remain as they were given, but we learn to know them and their functions 
 better, and to use them practically in our art without feeling shocked or 
 lost amongst the more refractory ones 2 . 
 
 1 So the distinctions made by such a writer as Johannes de Garlandia show rather a 
 keen sense, than a 'parade' (81, isa) of scientific accuracy. His series is,; unison, octave; 
 5, 4; 3, III; VI, 7; H, 6; 2, T, VII. Cf. 6, vol. i. 104 f. 
 
 2 It may be of some objective interest to note that I made a reference to Gaudentius 
 on pages 15, 108 above, but at that time I saw (like all other students of the subject, 
 I suppose) nothing specially significant in the term paraphony. In fact I had in the 
 meantime till writing the present section forgotten the existence of this term. The train 
 of thought expounded above made me feel the need for a term to cover the range between 
 consonance and dissonance, but different from either. Latin is here insufficient unless 
 we say sonance; so I put together the term paraphony. Months afterwards on turning 
 over the pages of Gevaert I was astonished and delighted to see the word paraphony 
 there. The point of interest is that whether unconscious cerebration was agog in this 
 coincidence which I greatly doubt or not, the objective differences analysed and dis- 
 cussed demanded the term; it was not suggested by, or transferred from, Gaudentius. 
 So perhaps ray analysis will do him a good turn for the dignity he gives to the term I 
 chanced upon.
 
 CHAPTER XIX 
 
 MELODY (OR PARAPHONY) AS THE PRIMARY BASIS OF MUSIC 
 
 WE have now reached a point of view from which we can survey a large 
 part of the realm of music. 
 
 The view we obtain has been clearly indicated in the thesis which 
 forms the title of this chapter. Melody is the primary basis of all music. 
 By melody we mean a special phenomenon of motion or passage between 
 two tones that appear before a mind in successive moments separated 
 from one another by a certain interval of time which may vary in 
 size within certain limits under various conditions. The successive tones 
 must not be so different (in loudness and blend) from one another as 
 to appear to come from different sources and so to suggest an objective 
 independence of one another. That circumstance is unfavourable to 
 melodic connexion. But it is not our present concern to study the 
 nature of the motion involved in melody, or in short of melody in 
 general. We have here taken melody for granted, as a familiar pheno- 
 menal fact. The reader is supposed to know already what melodic 
 connexion is, as he surely does, being able to tell at once whether the 
 ' passage ' from one note to another is there or whether there is a break, 
 a suspension of motion or passage, as there is for example after a close 
 or a half close, etc. (For an account of the primary theoretical study 
 of melody in this sense, see 77, chapter vi.) 
 
 Melody, in short, is the motion of music. 
 
 But the word is often used to mean more than that, namely the 
 series of pitches through which a melodic motion passes. For general 
 theoretical purposes it is best to use for this a word that links this 
 feature of sound to the analogous feature in the other senses. When a 
 visual motion passes through a number of points, we say it marks out 
 a certain form or figure. Thus a burning torch swung quickly round 
 leaves a trail that forms more or less of the circumference of a circle. 
 A flying meteor marks out a straight line, and so on. We may say 
 similarly that a melody falls into, or has, a certain form or figure. The 
 notion is quite familiar to musical literature. Perhaps the word ' theme ' 
 and its derivatives ' thematic ' and ' thematised ' are less open to confusion 
 with heterogeneous subjects than are any of the other words that bear 
 a similar meaning, such as tune, motive, subject, etc.
 
 CH. xix] THE PRIMARY BASIS OF MUSIC 161 
 
 If we use the term 'melody' for the general notion of motion from 
 note to note, then we may divide melodies into two classes those 
 that are thematic or that show a definite form capable of coherent 
 analysis, and those that are not thematic. In the latter there may be 
 plenty of motion from tone to tone, i.e. plenty of melody in general, 
 but as little form as there is in the motion of a fly to take a homely 
 instance. The fly is always on the move, and so is a dancer. But there 
 is figure or form in the dancer's actions, while in the fly's there is practically 
 none. 
 
 In this sense melody is the primary basis of all music. 
 
 With so general a meaning in the term melody, this statement 
 partakes very much of the nature of a truism. Music undoubtedly 
 began as melody. Apart from purely rhythmic art in which sound 
 plays only the part of a practically unvaried medium, all early music 
 is simply melodic. And it is thematic in a more or less simple way as 
 well. But it is generally supposed to have ceased to be wholly or 
 thoroughly melodic at a certain point of its development in Europe, and 
 to have become harmonic; and in so changing to have struck into a 
 new line of development that was present only in minute traces, if at 
 all, in primitive or in ancient music. This new line has led to wonderful 
 forms of art, unshadowed and undreamt in the first origins of music. 
 Harmony seems to be a new creation within music; a new dimension, 
 one might say ; the perpendicular complement to the horizontal functions 
 of melody, as has been said. 
 
 No doubt harmony has come to be of great importance. It is not 
 easy, however, to say precisely what its scope is. But there seems 
 little doubt that the melodic functions of music have been considerably 
 underestimated. In fact, from the point of view of theory, harmony 
 has usually been put down as the one and only basis of true music. 
 A music in which harmony is evidently neither implicit as in polyphony, 
 nor explicit as in harmonic music, hardly deserves the name of art. 
 It is merely primitive play, as it were. 
 
 An almost contrary thesis may be vigorously maintained. It may 
 be claimed that melody is the primary and continuous basis of music; 
 remaining so even throughout harmonic developments, which are 
 essentially a by-product of melodic complexity, always carefully sub- 
 ordinated to the prior and essential requirements of melodic movement. 
 
 In the earliest music there is only one line of melody, only one voice. 
 Harmony shows itself at most only in the reduplication of this melody 
 
 w F. H. 11
 
 162 MELODY (OR PARAPHONY) AS [CH. 
 
 at the octave, fifth, or fourth, or in an irregular accompaniment of 
 intervals to the tones of the melody which does not make a second 
 melody and does not seem to interfere with the apprehension of the 
 one distinct melody. From this somewhat chaotic state the art of 
 polyphony in two and then in three or more voices gradually emerges 
 in that the accompanying tones take on the form of a distinct voice 
 chiefly by the device of contrary motion with the chief voice 1 . Then 
 the art of polyphony becomes clearly conscious and slowly attains a 
 sense of the true principles of style suitable to simultaneous melodies 
 (cf. 48, chapter 11). The rules regarding consecutive and exposed 
 intervals are then gradually discovered. These rules are necessary in 
 order that the melodic distinction of the voices may be clear and easy. 
 The arrestive effect of the 'symphonies' is of the greatest importance 
 where arrest of melodic progression is desired; but it must be carefully 
 avoided when melodies have to flow easily together. Diaphonies must 
 not lead to the confusion of melodies; we must be led safely through 
 them and on to groupings of tones that the ear can readily apprehend. 
 But there is in this kind of art no difference, except that of difficulty, 
 between the construction of two simultaneous melodies and of three 
 or more. Harmony does not necessarily become explicit with the 
 grouping of three voices ; nor does music in two voices (now) necessarily 
 fail to be apprehended harmonically; there are many who claim that 
 even a single melody is necessarily apprehended harmonically, absurd 
 though that claim in the face of primitive music must be held to 
 be. Of course polyphony can hardly begin to be apart from a cogently 
 melodic or even thematic treatment of the combined melodies. That 
 follows naturally from their very being as distinct melodies. There 
 would be as little interest or beauty in the erratic or merely melodic 
 motion of several voices as there would be for us in a song made up of 
 a more or less random succession of tones 2 . 
 
 1 It was Helmholtz that suggested that "the first of such examples could scarcely 
 have been intended for more than musical tricks to amuse social meetings. It was a new 
 and amusing discovery that two totally independent melodies might be sung together 
 and yet sound well" (20, 244). But the actual course of development must have been 
 more natural and continuously meaningful than that. 
 
 2 C. V. Stanford (62, 6) extends this idea in a general rule with considerable emphasis: 
 "Mere combinations of notes, in themselves sounding well, but without logical connexion 
 with their successors, are useless as music. The simultaneous presentation of two melodies 
 which fit each other is at once a musical invention; and when a third or fourth melody 
 is added to the combination, the result is what is called harmony. To speak of studying 
 harmony and counterpoint is, therefore, to put the cart before the horse. It is counterpoint 
 which develops harmony and there is no such boundary wall between the two studies 
 as most students imagine." Our conclusions re-affirm this last statement.
 
 xix] THE PRIMARY BASIS OF MUSIC 163 
 
 Out of polyphony harmony was gradually abstracted by a slow 
 process of association of simultaneous tones and familiarisation with 
 their perpendicular aspects. The lines of melodic motion had to meet 
 often and often ere the patterns of their contacts could become 
 thoroughly known. And still longer time was required before men 
 found that these patterns could be ranged in succession in most fascinating 
 ways. This origin of harmony is in a sense quite familiar. It has been 
 well expressed by Sir Hubert Parry in his articles in Grove's Dictionary 
 of Music and Musicians. At the end of the article on Harmony he 
 speaks of it summarily as "the principle that harmony is the result 
 of combined melodies." 
 
 The ecclesiastical cadences were nominally denned by the progressions of the 
 individual voices, and the fact of their collectively giving the ordinary Dominant 
 Cadences in a large proportion of instances was not the result of principle, but in 
 point of fact an accident. The Dominant Harmonic Cadence is the passage of the 
 mass of the harmony of the Dominant into the mass of the Tonic " (46, sio). 
 
 But harmonic music has not ceased to be essentially melodic. The 
 thematisation characteristic of polyphony has disappeared from the 
 majority of the voices perhaps; but each chord must still be connected 
 with the preceding in ways which make melodic connexion easy and 
 which do not allow its clear flow to be arrested by excessive symphony 
 or dissipated by extreme diaphony. The paraphonic components of 
 chords are perhaps their essential musical constituents. And the 
 discovery of new chords may be said to be the development of new 
 paraphonic combinations. Most of the modern systems of chord forma- 
 tion have emphasised the importance of the superposition of thirds as 
 a principle of origin. We can now understand it not merely as an 
 empirical principle, but as a principle of function (in so far, of course, 
 as it rests not upon a mere arithmetic of thirds by transposition through 
 octaves, but upon a naturally felt, and so real, connexion through the 
 similarities of inversions to one another as patterns). The primary need 
 is not so much a wealth of consonantal and dissonantal effects, as a 
 clear and intelligible flow of melody through such harmonic effects as 
 are compatible therewith. 
 
 "By the use of chromatic passing and preliminary notes," Sir Hubert Parry 
 says, "by retardations, and by simple chromatic alterations of the notes of chords 
 according to their melodic significance, combinations are arrived at such as puzzled 
 and do continue to puzzle theorists who regard harmony as so many unchangeable 
 lumps of chords which cannot be admitted in music unless a fundamental bass 
 can be found for them*' (46, sis; cf. 47, and 48, ss*). 
 
 112
 
 164 MELODY (OR PARAPHONY) AS [CH. 
 
 This interpretation of the development of chords is to be preferred 
 to Stumpf's. It is certainly not necessary to introduce the notion of 
 concordance in order merely to explain the conjunction of more voices 
 than two. The notions of consonance, or better, of paraphony as may 
 perhaps be said generically are quite adequate to the use of intervals, 
 not only between two voices, but also between more than two. No 
 change of basis or of principle is thereby required. The essential basis 
 of music ever was and remains melodic movement. One melody is 
 self-contained; simultaneous melodies must be mutually compatible. 
 Single melodies are almost inevitably thematic; simultaneous melodies 
 may be so too; but the thematisation of one or more may be dropped 
 in favour of the interest created by their harmonic fusion. Or, finally, 
 as D. F. Tovey expresses it, modern melody may be merely (is) "the 
 surface of a series of harmonies" (75) 1 . When harmony emerges, 
 however, it is not a new creation; it is still precisely the same thing 
 as is the fusion or balance of two tones, except, of course, that it is 
 of greater scope and detail in several voices than in two. It has now 
 only been made the centre and object of the artist's creative genius. 
 The art has become perpendicular in build instead of horizontal. 
 
 But the rejection of harmony or concordance as an essentially new 
 element in polyphony does not imply that the latter in its harmonic 
 form has brought forth nothing. There has certainly been great develop- 
 ment and growth, so much indeed as to create striking differences in 
 the styles of art. The interest created by these was so great that much 
 of what had been toilfully gained in polyphonic art was temporarily 
 abandoned and fell into common neglect. That must perhaps always 
 happen to any art when new constructive vistas appear. 
 
 There seems to be no doubt that of the acquisitions of the new 
 art the most important and fundamental was the principle of tonality. 
 In polyphony the thematisation of all the melodies held them artistically 
 together. This common structure could only be abandoned when a new 
 principle of connexion had become apparent, something that would 
 link the various voices together throughout the changes of their harmonic 
 patterns. These patterns had to be wrought into an intelligible system, 
 and, as we know, even the outline of the scales had to be altered to 
 make this possible. 
 
 1 Or, "modern melody is the musical surface of rhythm, harmony, form, and instru- 
 mentation. In short melody is the surface of music." This may be the modern culmination 
 of melody, but it is certainly a wrong definition of melody in general, including primitive 
 melody.
 
 xix] THE PRIMARY BASIS OF MUSIC 165 
 
 It is tonality that gave and continues to give the chief impulse to 
 the systematisation of chords. These are, and perhaps can only be, 
 systematised in relation to tonality, and its three poles tonic, dominant 
 and subdominant. It is a perversion of actual history to suppose that 
 chords were first derived from the mere tone (fundamental and partials), 
 bringing implicit in them the determinants of tonality and of our 
 scales. Scales were developed long before either tonality or chords 
 had been conceived or even felt. On the contrary the course portrayed 
 in history seems systematically much more acceptable. First came mere 
 motion in tones (without any implicit scale) or mere melody; out of 
 melody was begotten by the force of familiarity and the needs of social 
 co-operation scale; scale made further complications and co-operations 
 possible, leading to polyphony; the habits and traditions of polyphony 
 engendered harmony and tonality; and that as it grew reflected upon 
 its progenitors and moulded them to its better development; it adjusted 
 the scale to its systematic requirements and reduced the functions of 
 thematised melody in favour of shorter bonds of melody running 
 through and combining large tonal masses. Thus line was reduced in 
 favour of mass; but the mass itself is now also treated linearly, as it 
 were; or it is made to move round a central axis of orientation. It is 
 only after this time that we find the first beginning of a systematic 
 exposition of chords in Rameau. And we must remember that, 
 curiously enough, one of the first steps in this work was the identification 
 of those groups of tones that we know as inversions of one another. 
 
 We have already discussed the problem of inversions and have 
 introduced the notion of ' volumic pattern ' to account for their connexions 
 and differences. Possibly the chordal aspect of modern music is founded 
 upon this attitude towards volume, whereby the group of tones is con- 
 sidered rather as a whole than as so many stages in so many melodies 
 of which the themes are clearly held in mind. Of course the mind has 
 still to be led melodically through these masses. But the interest lies 
 not in the forms created by the melodic movements as such, but in the 
 masses, their patterns or ' surfaces ' or ' colours ' (as some say) into which 
 the mind has been easily and safely led. The modern interest would lie, 
 then, not so much in the forms of all the motions through which the 
 listener is carried, as in the phases or masses of sound in which he from 
 moment to moment finds himself. 
 
 The canvas is now filled with broader effects. These are still aD 
 built upon lines of movement (i.e. melodies), but the lines are now 
 subdued and hardly appear from a distance, so to speak, except it
 
 166 MELODY (OR PARAPHONY) AS [CH. 
 
 may be in the one, or perhaps two, that run through the mass in its 
 changes and give its movements thematic form and beauty. 
 
 Tonality may be said to be perhaps the broadest of these effects. 
 In its earliest forms it is merely a centre or point for the harmonic 
 stream. In its later and freer developments it might almost be considered 
 as a new form of movement a new melody of all the (unthematised) 
 voices at once and it builds itself up like the primitive scale upon the 
 simplest consonances, of which the fifth is, of course, the first distin- 
 guishable one, the octave being by its function already mere identity 
 or repetition. So we get the poles of tonality, the dominant and 
 subdominant. And these again give others, equally closely allied to 
 them, until a whole system appears through which the harmonic stream 
 may be made to wander for its greater diversity and beauty. 
 
 One of the perennial problems of tonality has been the nature and 
 origin of the difference between the major and the minor scales. Their 
 actual constituents, as we have noted, have been determined by the 
 systematic requirements of the groupings of the moving voices that 
 are most essential to them. In their most rudimentary form these 
 reduce entirely to the two thirds (and sixths) the two chief inherently 
 paraphonic intervals. The tonal difference between the major and 
 minor keys may then be read from their symbols of 'origin' in the 
 two common chords ceg and ce^g. 
 
 The different relations of these two to the partials of their supposed 
 root c have been grossly overrated in importance. Certainly, if c has a 
 full series of lower partials, e and g will coincide with them. But, as 
 has so often been pointed out, the partials of e and g, which cannot 
 be ignored, confuse the issue. For e gives g$ and b, while g gives 
 b and d. As Macfarren said : 
 
 It is of course necessary for practical musical purposes, not only to make a 
 selection of notes from the endless harmonic series, but to confine the use of harmonics 
 to those belonging to certain exceptional generators, or roots, in every key ; otherwise 
 every note of every chord might be supposed to furnish its harmonic series, and 
 each of these its harmonics in turn, all sounds would confuse all other sounds, 
 tonality would be at an end, and Babel would reign supreme (35, 94). 
 
 In spite of the fancies of recent composers in search of new scales 
 (23, 72 f., and passim; cf. 5, 24), these harmonic origins have for 
 many a long year obviously been as dead as a door-nail. A much 
 simpler origin of such latest varieties is their actual source the chro- 
 matic scale of just or of equal temperament, consciously treated by the
 
 xix] THE PRIMARY BASIS OF MUSIC 167 
 
 process of 'aesthetic selection' that Helmholtz emphasised. No doubt, 
 however, the coincidence of partials will make for slight differences 
 of smoothness; these we have already admitted as modifications of 
 smoothness otherwise given. 
 
 Paraphonically the two common chords show practically no difference 
 at all. Both contain a major and a minor third and a fifth. The major 
 third is 'exposed' in the major chord by its resting on the bass voice; 
 the minor third is similarly 'exposed' in the other. Thus the two 
 chords are practically equivalent. 
 
 The only difference that remains is the difference of pitch of the 
 middle tone. It lies lower in the minor chord than in the major. And 
 this difference must run equally throughout the whole of the two scales. 
 Any composition distorted from its major tonality into the tonic minor 
 differs from the former solely in the lowering of all the mediants and 
 submediants by a semitone. The minor form differs from the major 
 by an inner lowering of pitch. It is more 'voluminous,' heavier, darker, 
 sadder, etc. Change to the major key means a streak of lesser volumes, 
 a brighter, lighter, clearer atmosphere, as it were 1 . 
 
 When the problem is thus cleared of its false and artificial difficulties, 
 the solution is easy and it appears to be quite natural and inevitable. 
 There is no special agony of harmonic birth in the minor; we do not 
 look upon the major from below and upon the minor from above; 
 they are not the mirror images of one another. The minor tonality 
 partakes merely of that difference that appears in the raising of the 
 pitch niveau of a composition. Smaller volumes suggest brighter, 
 lighter effects; or they merely are smaller, more precise, tones, apart 
 from all suggestion. And so the minor key appears woven with one 
 larger thread throughout ; our minds respond to this auditory difference 
 with any analogously varied experiences we may have ready in our 
 memories and moods 2 . 
 
 It is a familiar fact that in recent years many experiments have 
 
 1 Cf. the detailed empirical investigation of Becker's, where it is shown that "Dur 
 bedeutet schlechthin Lust, moll schlechthin Unlust" (2, aw); and (p. 258): "As expressions 
 of pleasure in the minor are not free of unpleasant moments, while on the contrary major 
 as expression of displeasure approximates by a greater or less indefiniteness of coloration 
 to the minor character, and, besides, the displeasure value of many contents is not very 
 distinct, therefore we can uphold the meaning of the modes as opposite feeling-tones of 
 corresponding musical devices." 
 
 * Contrariwise and similarly, for the ancient Greeks "the passage of the minor third 
 into the major expressed a lowering, a depression" (16, 324). For in their music the melody 
 lay below the accompaniment, so that the major third (reckoned downwards), gave the 
 touch of weight or sadness. On thirds in Greek music see also 10, aw.
 
 168 MELODY (OR PARAPHONY) [CH. xix 
 
 been made by composers towards the establishment of new schemes 
 of melody and harmony. It is unnecessary to specify or to describe 
 these here. Time must first show what is their capacity for lasting 
 artistic usage. But it is evident that the basis of music expounded in 
 the preceding chapters is quite ready and able to accommodate all that 
 is proved to be acceptable in them. If our theoretical analyses and 
 constructions correspond fully to the chief long tried and approved 
 forms of music, they will apply to the newer departures in their degree 
 of success. For these actually hold already for the musical mind a 
 very definite relation to the earlier established music. We hear and 
 know what we gain in the new and what we lose in it that made the 
 old precious. There is in all a gradual transition and development. 
 The mind may grow familiar with the natural and artistically achieved 
 paraphonies of the main drift of musical art and tire of them. It may 
 then strive to curb the lesser paraphonies to its will. But the gain of 
 novelty in chords of fourths must involve a great strain upon the 
 apprehension. Many groupings of lesser paraphonies must be in actual 
 effect much more diaphonic than paraphonic, discordant more than 
 melodious. But it is not our present intention to defend or to justify 
 any one of such experiments, but merely to show that the basis of 
 analysis and theory already offered not only leaves room for them, 
 but can even anticipate the losses and deficiencies they are likely to 
 entail. That, as already said, can be done without any theory, merely 
 from the analytic foundations of harmony, as it is generally known. 
 The theory given above merely shows how these effects are based in 
 the actual auditory stuff itself. 
 
 Of course, only the fringe of this vast subject has been touched. 
 One of the great difficulties that face the theory of music in general at 
 the present time is that the art is so highly developed, while in spite 
 of the vast amount of analytic work that has been done in connexion 
 with the form and structure of music the theory of the basis of music 
 in the auditory stuff of. tones has hitherto been really non-existent. 
 Reaching the fringe of the subject, therefore, implies much more than 
 at first appears. We have not yet explained much. For the detail of 
 that work the centuries remain. But we have at least now a fairly 
 clear view of the promised land, in search of which men have wandered 
 so widely and aimlessly.
 
 CHAPTER XX 
 
 THE FACTORS THAT MODIFY PARAPHONY 
 
 WE may now use the term paraphony to indicate not only the specifically 
 paraphonic intervals of thirds and sixths, but also all intervals 
 symphonic as well as diaphonic, specially consonant or dissonant, 
 in so far as they are made, or become, paraphonic, or in so far as their 
 actual or potential paraphonic function is capable of modification by 
 various factors. These factors we have already to some extent 
 encountered in the previous chapters, and shall now proceed to gather 
 together and to resolve as far as possible into their essential functions. 
 We have not only to indicate the effect of each factor, but as far as 
 possible also to explain how that effect is attained. 
 
 We notice, then, that paraphony diminishes from its optimum in 
 the thirds and sixths in two directions : in the one towards the sym- 
 phonies, which create a loss of distinction in unity and thereby a point 
 of relative balance and rest of tonal mass; in the other towards the 
 diaphonies, which make a loss of distinction in confusion and thereby 
 a point of relative restlessness and propulsion of the tonal mass. 
 
 We are also familiar with the nature and ground of the varied 
 exposure or ease of distinction of the various pairs of voices : in the 
 following decreasing order, B-S, B-A and B-T, S-A and S-T, and 
 A-T (cf. above, p. 102 f.). 
 
 The effect of an increase in the number of voices is largely due to 
 the mere spreading of the attention. It is more or less a general rule 
 of sensory apprehension that the larger the number of distinct items 
 to be observed simultaneously, the less distinct is each one, and the 
 less easily is it separated from the others and observed, especially in 
 the field of hearing, where fusion and overlapping of volumes play so 
 important a part. The various pairs of voices will, of course, still have 
 the same relative grades of exposure, the outer voices being the most 
 exposed and the innermost ones the least so. But as the maximum 
 grade will now be lower than it was, the minimum will be doubly so; 
 for the larger number of voices makes a larger number of grades. 
 Probably the difference between these grades is also smaller. Thus 
 none of the melodies will be quite so cogent in general, unless one or
 
 170 THE FACTORS THAT MODIFY PARAPHONY [CH. 
 
 other is made specially prominent by increased intensity or by specially 
 distinctive or obtrusive tone-blend (as for example in orchestral music). 
 The effect of symphonies or diaphonies upon the flow of melody will 
 also be less marked. Any slight disturbance is hardly noticeable in 
 the mass. 
 
 Besides any symphony or diaphony is now likely to be separated 
 by several voices, so that the unity that might otherwise be arrestive 
 will now be variegated, and the clashing that might be confusing will 
 be well spread out and manageable. Even if the thread of two voices 
 is temporarily lost, no great risk will be incurred; for there is much 
 else to engage the attention, and the harmonic patterns of the whole 
 chordal masses will still prevail. 
 
 Contrariwise, when the number of parts is decreased, what is allowable 
 in four parts will be governed by apparently stricter rules. The decrease 
 in the number of voices makes each more prominent, so that the rules 
 for it partake more of the stringency of the rules for outer voices. At 
 the same time the innermost voices fall away, and, as the rules for 
 these are the most lax of all, the general average of strictness is apparently 
 raised. 
 
 The next question is one of the most important and very puzzling 
 in its own way, unless it is faced with the resolution of rigorous logic : 
 what is the precise nature and basis of the effects of similar and contrary 
 motion? 
 
 Oblique motion must, of course, not be forgotten. But the answer 
 for it is obvious : one of the voices stands still and is thus already 
 melodically prepared for the attention, so that the other moving voice 
 has the greater freedom and scope so far as the analytic attention is 
 concerned. In other words the melodic spontaneity or force of the 
 moving voice is a matter of its own coherence and expressiveness; 
 and thus the melodically blurring effect of any symphonies and 
 diaphonies that may occur between the two voices is obviated. The 
 early use of oblique motion in connexion with dissonances is familiar. 
 Similarly we have noticed how an otherwise unstable melody has more 
 cogency, and may even progress by leap, if the other tones in the chord 
 remain unchanged, or if the two successive chords differ only in position. 
 
 With regard to similar motion, it is exceedingly tempting to interpret 
 it as a positive influence. Its treatment in all textbooks of harmony 
 suggests this strongly. The recent notion of 'exposure' of octaves 
 and fifths (58, 268 f.) has greatly increased the force of this suggestion.
 
 xx] THE FACTORS THAT MODIFY PARAPHONY 171 
 
 But that indication seems to be very misleading. For similar motion 
 has to be kept in relation with aD the intervals. And it is important 
 to notice that it has no apparent effect upon paraphonies at all. This 
 would compel us to infer that its effect upon the symphonies and 
 diaphonies is also nought; it merely allows them to stand untouched, 
 relieving neither voice of the necessity for being cogently melodic and 
 paraphonic. This the diaphonies and still more the symphonies fail to 
 be ; for while the former urge us on in spite of the confusion they tend to 
 produce, the latter arrest the melodic flow a most pernicious effect when 
 it is sufficiently ' exposed ' by the prominence of the voices that bear it. 
 
 The difference between one symphony or diaphony and consecutives 
 is then simply the difference between one bad effect and two of them in 
 succession 1 . One has only to listen to consecutive fifths for a while to 
 notice that the bad effect that appears in them comes out not merely 
 and solely when a second one is played, but that it attaches even to a 
 single one. The fifth, and to a less extent the fourth, is a bad interval 
 for polyphony in general, i.e. for paraphony. No doubt it sounds well 
 as a merely momentary or isolated mass of sound, or so long as we 
 think of it as detached from all melodic flow or sequence. But it is 
 commonly recognised to be (otherwise) bare and poor. The octave will, 
 of course, be still worse as an interval than is the fifth, in so far as it 
 is heard as an interval. That happens really only in polyphony. When 
 it is played alone, we tend rather to apprehend it in its (then best) 
 musical function, merely as a reinforced single tone. 
 
 Similar motion, therefore, is a negative condition. The positive 
 force must rather be contrary motion. That will have the effect of 
 favouring in all cases a distinction of the voices, a distinction both 
 in symphonies and in diaphonies and a greater distinguishability in 
 paraphonies. Contrary motion, in other words, favours the paraphonic 
 effect all round. 
 
 The only variant on these conclusions that might be offered is the 
 assumption of a certain degree of blurring and confusing effect in the 
 case of similar motion. The degree, however, would have to be small 
 enough not to affect the thirds and sixths disadvantageously. This 
 modification would still leave the theory on the whole identical with 
 the inferences stated above. In both forms similar motion would rank 
 as a relatively negative factor. 
 
 1 This is suggested by Shinn (58, as?) in one instance when he says: "The fact that 
 we admit consecutive fifths when so formed [in connexion with discords], obviously 
 covers the admission of exposed fifths similarly formed."
 
 172 THE FACTORS THAT MODIFY PARAPHONY [OH. 
 
 It may seem to be an enigma in this connexion how it comes that 
 the relations of motion towards a chord that is not yet sounded makes 
 the analysis of that chord easier or harder, seeing that there is no 
 continuity or slur of sound between this chord and the previous one. 
 The objection is well founded. The second chord falls upon us unawares 
 (more or less); many other chords might usually have been played 
 instead. The process, however, is grounded in the nature of melody 
 (in the most general sense) as a motion or phenomenon of motion from 
 one tone to another. A second tone is a sort of reservoir or line of 
 drainage, into which the residual (neural) activity of the first tone 
 runs off and discharges, so that the two become linked together by a 
 line (as it were) of activity which can only emerge when the second 
 tone has made a place for the first to discharge into. Paraphonic 
 differences rest upon the fact that these discharges can be fully controlled 
 only under certain circumstances. And the artistic use of melody 
 requires their full control. 
 
 In contrary motion, then, the tones of the second interval both lie 
 ordinally outside or inside the pitch-range of those of the first. The 
 motions between the respective pairs of tones are, therefore, easily 
 distinguishable, and so we get a paraphony the flowing of two melodies 
 side by side. In the case of successive thirds and sixths this paraphony 
 is quite natural. The tones of these intervals are evidently just the 
 right distance apart for easy melodic flow towards or away from them- 
 selves, even though another third or sixth follows. Similar motion does 
 not spoil this effect at all. So in symphonies and diaphonies the bad 
 flow of melody is due to the nature of the intervals, whereby their 
 tones enter into confusion with one another ; it is not due to the similarity 
 of motion, at least for the greater part. 
 
 We can now see readily the basis of the familiar rules that (melodic) 
 parts should not overlap or cross one another. If they do, there is a 
 great probability of a confusion of melodic connexions and so of faulty 
 paraphony. The same applies to such rules as : "If the same note is 
 found in two consecutive chords, it should in general be kept in the 
 same voice, ... as it will conduce greatly to the smoothness of the 
 part- writing"; and: "Each part should generally go to its nearest note 
 in the following chord" (52, 40) 1 . But if a difference of tone-blend is 
 given, as in choral or chamber music, the voices may move more freely ; 
 
 1 Cf. a more "modern" expression (51, M): "There must be a connecting link 
 between successive chords of a note common to each or of one or more parts moving within 
 the interval of a tone or a semitone, the other parts being free in movement."
 
 xx] THE FACTORS THAT MODIFY PAKAPHONY 173 
 
 for the blend of each will serve to bind its component tones together 
 (cf. p. Ill, above). It is also apparent why a second should not proceed 
 to a unison; for, without the distinguishing help of contrary motion, 
 this is just confusion worse confounded. In two successive unisons 
 also we tend to lose sight of the intended duality of parts. But one 
 unison is harmless, since the parts must proceed to it from a good 
 paraphony and by contrary motion, while, in leaving it, contrary 
 motion will again be very frequent or an easy paraphony the objective. 
 Thus the two voices drop easily into the one reservoir and as easily 
 discharge towards the following two. The one tone is then really a 
 unison psychically. Progression to a unison by similar motion, however, 
 involves the crossing of parts and is only tolerable under favourable 
 circumstances, as by step in one part, or from dominant to tonic, and 
 the like (cf. 52, 31). Relations of visual motion very like these motions 
 of tone have been experimentally established (78). 
 
 The interests of harmony are specially concerned in the next funda- 
 mental problem : what is the effect of the paraphony of simultaneous 
 intervals upon one another? Or, how do they combine to a resultant? 
 An analogous question emerged naturally in the development of the 
 study of fusion (as the approximation of two tones to the impression 
 made by a single tone), viz.: what degree of fusion appertains to an 
 assemblage of three tones, and how do the individual fusions of its 
 three intervals contribute to the result? But, whatever may be the 
 value of that enquiry in itself, we have already seen that it is not the 
 proper line of approach towards music. The interval is, of course, in 
 a sense the element of structure in music. But our systematic inductions 
 have shown us that the decisive consideration in the function of each 
 interval is how the two melodies of which it forms a phase of conjunction 
 flow together through it. The analysis of music naturally reduces the 
 problem in the first instance to a study of pairs of melodies. For the 
 least grade of harm is the mutual disturbance of two melodies. But 
 there is no reason why three or more melodies should not disturb 
 each other. And music may properly consider these interferences if 
 they occur in typical forms. Our problem is merely an extension of 
 our study from two concurrent melodies to three or more. 
 
 But in thus declining the implications of the theory of fusion and 
 in denying its capacity for progress towards a proper theory of music, 
 we do not bind ourselves to reject the serial arrangement of the intervals 
 in respect of something which makes the one preferable to the other
 
 174 THE FACTORS THAT MODIFY PARAPHONY [OH. 
 
 (call it 'greater consonance,' if you like) upon which the theory of 
 fusion is based. In that arrangement the major third stands before the 
 minor third and the major sixth before the minor sixth. We shall, 
 however, in due course have to look back upon the theory of fusion or 
 of paraphony or of consonance, or whatever we may call it, and attempt 
 to find a sufficient basis for the functions of which we are now gathering 
 a fuller and completer knowledge. 
 
 In approaching this problem of summation, therefore, we must 
 make clear to ourselves first of all what it is that may be added together. 
 In symphony there is a loss of distinction in unity and a consequent 
 arrest of melodic movement. The effect of this loss may, as we have 
 seen, be annulled by the use of contrary motion and other devices. 
 But the unity of the symphony survives to characterise the moment of 
 conjunction of the melodies and to give it an aspect of unity, steadiness, 
 and stability, which must not be heightened if the music is to move 
 smoothly. In the paraphony there is freedom of distinction and of 
 melodic movement. In diaphony there is a loss of distinction in confusion 
 which again may be relieved, as far as approach to it is concerned, by 
 special devices of an analytical tendency. But these of course do not 
 annul the basis of confusion in the stuff of the tones themselves, whatever 
 it may be. They only make the conjunction melodically serviceable 
 and clear in spite of its confusion. Or perhaps it keeps in a state of 
 tension and incompatibility what would otherwise be open confusion. 
 
 Thus we see that we can hardly expect to find it possible to add 
 symphonies and diaphonies to a resultant, as we add positive and 
 negative quantities together. But there seems to be no ground of 
 incompatibility between either of these and the neutral paraph onies. 
 And we may safely infer that the conjunction of several symphonies 
 or of several diaphonies will produce an effect of greater unitariness 
 and arrest or of greater tension and harshness. 
 
 Thus in ceg there are two paraphonies ce and eg and one symphony eg. 
 The general character of the chord is, of course, symphonic, i.e. it is 
 paraphony bound together by symphony. If the c 1 above is also given, 
 there are added a paraphony ec l and two symphonies gc l and cc 1 . The 
 octave displaces the fifth in the outer voices and the fourth appears 
 in a relation of medium obscurity. Thus on the whole the symphonic 
 effect is increased. In the chord bdf 1 there are two (minor third) para- 
 phonies, and one diaphony bf 1 , so that the general character of the whole 
 is diaphonic. 
 
 But the different exposure given by the different pairs of voices
 
 xx] THE FACTORS THAT MODIFY PARAPHONY 175 
 
 will make differences between chords that consist of the same intervals. 
 We should expect the symphonic (or diaphonic) effect to be greatest 
 when the greater symphony (or diaphony) lies in the outer voices, 
 less when it lies between the soprano and an inner voice than between 
 the latter and the bass 1 , and least when it lies in the inner voices. Thus 
 ceg is a greater symphony then cej?</, because the major third lies in 
 the former on the bass, while in the latter it lies on the highest voice. 
 Of the inversions of the chord bd 1 / 1 , bd 1 / 1 is the worst because the tritone 
 lies in the outer voices, fbd 1 is next because it now lies on the bass, 
 and dfb the best because here it is obscured in the upper voices. The 
 arrangement dbf 1 is even better; for, while the tritone remains in the 
 same place, the minor third now displaces the major sixth in the outer 
 voices, and so increases the tendency towards symphony somewhat 2 . 
 
 Similarly, of the inversions, ceg is the best, because it contains the 
 two best fusing paraphonies and has the fifth in the outer voices; 
 gcP-e 1 , the second inversion, is the next best, and egd-, the first inversion, 
 the least good, because the latter has not only lesser fusion for each 
 paraphony (3 for III, 6 for VI), but it also has its consonance the 
 fourth in the upper voices. In the minor chord the relations are 
 similar for the fourth, but the other two intervals are major in the 
 first inversion and minor in the second. How these sets of changes 
 balance out, is a fine point for experiment to settle. Kemp's experiments 
 show a balance of preferences with respect to the comparative fusions 3 
 of the two inversions of the minor chord, but a preponderance of 
 preference for the second inversion of the major (29, 207). 
 
 Regarding the alternative arrangements for each inversion we may 
 infer, in harmony with musical practice, that, when other things are 
 equal, the closer position is better than the extended one. Thus ceg is 
 better than cge 1 , because in the latter the fifth is not in the outer voices, 
 but on the bass. But in ge^c z as against gee 1 we get a minor sixth for 
 a major third, and an eleventh (now in the outer voices) for a fourth. 
 There is the same loosening of relations in ec 1 ^ 1 as against egc 1 . But 
 
 1 Cf. the rule stated by Kulpe and confirmed by Pear's experiments that where the 
 same intervals go to form different chords, the fusional degree of the chord is greatest 
 when the better fusing interval lies lower (29, 209). Cf. 21, Pt 2, 91 : "The more 
 perfect concords ought to be below, and the less perfect above, in a chord." 
 
 2 Cf. 35, : "In this inverted form it is classed among the concords of the ancient 
 style." 
 
 * This is Kemp's term for the object of preference in his experiments. We should 
 not admit the implications of the term in this connexion, of course, as has already been 
 sufficiently indicated.
 
 176 THE FACTORS THAT MODIFY PAR APHONY [OH. 
 
 the differences in these cases cannot be great, since the changes do 
 not produce any bad effects, while the connexions of pattern established 
 between original inversions and their alternative arrangements help to 
 bind them together again. Nevertheless the differences are certainly 
 there, and they call for experimental as well as for theoretical 
 study. 
 
 Where diaphonies appear in chords we find in practice that "the 
 lowest fusion contained in the chord is most decisive " (29, 209, 244). That 
 chord is most diaphonic which contains the greatest diaphony. The 
 more exposed it is, of course, the worse will be the effect, as we have 
 already noted and exemplified. 
 
 The greater decisiveness of the diaphony appears again in such 
 chords as the dominant seventh. But while the fifth does not outweigh 
 the two discordant intervals so as to render the whole chord purely 
 paraphonic and not at all diaphonic, yet, as we have seen, the discords 
 do annul the symphonic effect of the fifth. The fifth is still a fifth, 
 both as interval and as 'fusion'; but the arresting confusing influence 
 it exerts upon the two streams of melody that run through it, is now 
 annulled by the presence of the diaphonic intervals in the chord, 
 especially if the fifth is not isolated in the bass or soprano. So both 
 one fifth or a second fifth may pass unguarded in the body of such a 
 discord. 
 
 The 'root' position gbcPf 1 is better than fgbd 1 , because here the fifth 
 rests on the upper voice. In the other two inversions the fifth becomes 
 a fourth. Thus we see that the 'root' positions of chords are not at 
 all due to their derivation from the bass note by any indirect process. 
 The ground of preference lies in the chord itself, not in the supposed 
 partials of one of its notes. If the term 'root position' is to be retained, 
 let it be understood in the sense that in it the chord is, as we hear it, 
 stablest and most nearly symphonic. Moreover, as this stability is 
 referred naturally to the central pitch of the whole chord, i.e. to the 
 bass, the root position of a chord will form a specially good approach 
 to any difficulty in the bass melody, e.g. to a fourth on the bass. 
 
 It is a notable fact that both observation and general theory thus 
 place the second inversion of the major chord next to the root position, 
 preferring it to the first inversion. There is no escape from this theoretical 
 conclusion so long as the fourth is ranked as the third grade of con- 
 sonance. Even Helmholtz, whose theoretical foundations were the 
 successive steps of the harmonic series, was inevitably led to the same
 
 xx] THE FACTORS THAT MODIFY PARAPHONY 177 
 
 result 1 (20, 214). The verdict of experimental observation can be 
 challenged only on the ground of misdirection of description. But it 
 cannot be denied that a certain point of view leads to the ranking of 
 the second inversion after the root position and before the first inversion. 
 
 But while the musical observations of all time have readily admitted 
 the ranking of the fourth as a consonance immediately after the fifth, 
 the verdict of all polyphonic music has been equally in favour of the 
 inversion in which the fourth is in the upper voices and against the 
 second inversion. In fact the latter conclusion has threatened at times 
 to swamp the other, though it has never really succeeded in doing so 
 completely. The fourth obviously cannot be a dissonance generally, 
 since it is clearly consonant apart from the bass voice. Some special 
 circumstance must be responsible for the bad effect in the bass. 
 
 We have already (p. 135 ff.) discussed this question and have noted 
 that its probable basis is the proximity of the (major) third. The fourth 
 somehow suggests this other interval so strongly that it seems itself 
 to be only a point of transition to the third. In thus seeming to call 
 for a resolution, the fourth indeed resembles the dissonances. But the 
 resemblance is only accidental, in the logical sense. We know now that 
 the essential feature of dissonance is not merely its low grade of fusion, 
 for the sixths have also a low grade; a dissonance shows also a loss of 
 distinction in confusion of the two tones that compose it or of the two 
 melodies that pass through it. For the fourth, however, we cannot 
 claim any lower grade of fusion than that appertaining to the fourth 
 in general; and, even if there is a certain confusion in the bass fourth, 
 it is not an internal confusion that produces merely a loss of distinction, 
 a mere blurring of what is given in the interval; it is a confusion that 
 is due to the attraction of the neighbouring third; the confusion has 
 an external reference beyond the interval actually given. 
 
 It is plain to ordinary musical observation that a fourth, exposed 
 not only by standing upon the bass, but rhythmically as well, calls 
 for the third on the same bass, and that, if this call is to be suppressed, 
 
 1 Helmholtz referred the effect of the bass fourth to the disturbing effect of tonality. 
 Bat the tonic does not necessarily come into question at all. On the contrary the six-four 
 chord earliest and oftenest admitted is that of the common chord on the tonic, next is 
 that on the subdominant, and then that on the dominant (35,86f. ; cf. Prout, p. 133 f., above). 
 On the other degrees of the scale it is rarely used. Thus one might even claim that, far 
 from being disturbed, the chord is perhaps rather made tolerable by the influence of a 
 distinctly tonic reference. This would be confirmed by the fact that the peculiar character 
 of the six-four chord was recognised long before tonality had come clearly to the surface 
 of the musical consciousness. In fact, its character was then more stringently unique 
 than later. 
 
 w. F. M. 12
 
 178 THE FACTORS THAT MODIFY PAR APHONY [CH. 
 
 not only must the rhythmical exposure 1 be avoided, but a cogent melodic 
 line must also be driven through the bass note. And at least a great 
 part of the necessity for also approaching a bass fourth in a special 
 way is likewise due to this proximity of the major third. If the question 
 be raised how this third can possibly disturb the melodic flow of the 
 bass when even the fourth itself is not yet sounded, we must answer 
 by drawing attention again to the fact that a melodic point that is 
 about to follow upon one just sounded, provides a sort of outlet for 
 the latter's residual energy. If the second point lies near the first, the 
 discharge is easy and cogent. The further apart the two points He, the 
 less forceful is the transference. A melodic leap is in itself a difficulty. 
 But this difficulty can be increased by various circumstances : by the 
 unfamiliarity of the leap to be taken, and especially by the proximity 
 of an easier more consonant, more paraphonic, or more familiar 
 interval. The difficulty of the fourth from the bass is probably due to 
 this latter circumstance. The fourth is more consonant, it is true; 
 but the (major) third as the most consonant (i.e. best fused) paraphony 
 is both very important and very interesting. And the intonation of 
 a bass fourth brings any slender (leaping) melodic line so close to this 
 attractive third that the actual fourth will come as a sort of jar upon 
 the expectation. The melodic flow will be disturbed. Hence the necessity 
 of making the melodic line leading to the bass note cogent by the devices 
 summarised above (p. 134 f.). 
 
 When the bass note is firmly established, the third suggested is 
 that upon it. When it is not well established, the third suggested must 
 be that below the upper note of the fourth. This is strongly indicated 
 by the effort all the rules make to improve the cogency of the bass 
 melody. As the bass is the most exposed voice of all, any vacillation 
 in its course will make the whole chord unstable. 
 
 But there can be no such instability when the fourth is not on the 
 bass. When, as in cegc 1 , it lies in the upper voices, all the rest of the 
 
 1 Rhythmical exposure is of the greatest importance in music generally in so far as 
 it increases or decreases the good or bad, desired or undesired, feature of any purely tonal 
 element. The arrestive effect of the last chord of a cadence, for example, is heightened 
 by such exposure, as it must be if the 'cadential' effect is to be attained. It is at such 
 points that one sees the continuity between the 'dynamics' of tone and of rhythm a 
 large subject which deserves special treatment. But that can only be given when we 
 have brought the dynamics of tone into acceptable order. The dynamics of rhythm are 
 naturally more obvious and apparent. Among the ancient Greeks the dynamic of arrest 
 was induced largely by lowness of pitch. But the octave also appeared as a concluding 
 symphony (16, Index sub. 'Symphone,' 'Grave/ and 'Aigu'). Both of these effects are 
 still generally valid.
 
 xx] THE FACTORS THAT MODIFY PAR APHONY 179 
 
 chord, including its bass, calls ( for c 1 , which is at the same line a great 
 consonance, whereas b would give a great dissonance. Similarly in egc 1 
 the two paraphonies exposed upon the bass make c 1 easy, while 6 would 
 itself as leading note suggest the c 1 above it. 
 
 This course of reasoning strongly suggests that the basis of the bass 
 fourth anomaly must be merely one of a class of similar cases. Now 
 the anomaly of the fourth is due to its tendency towards the neighbouring 
 third. Hence we might express the principle of a class of such cases 
 as due to the proximity to one another in size of intervals of different 
 fusional or paraphonic character, and to a resulting tendency of the 
 one to point to, or to fall into, the other. 
 
 Methods of 'ear-training' make use of this feature of intervals as 
 well as of others. Thus the scheme propounded by Bridge and Sawyer 
 (3, 194) is as follows : Octave perfect unison of the two sounds so that 
 they sound almost as one; VII upper note will require to ascend; 
 7 both notes will require to move, upper down one degree, lower up 
 a fourth; VI upper will tend to fall to the dominant; 6 very strong 
 tendency to the dominant; 5 sounds bright, ear at rest; diminished 
 5 both notes will require to move towards one another by a second; 
 augmented 4 both will require to move away from one another by a 
 second; 4 upper requires to fall to the mediant; III calm, peaceful 
 effect, ear at rest; 3 peaceful, but more melancholy, ear at rest; 
 II close discord, lower note requires to descend a semitone; 2 a 
 fierce discord, lower note requires to descend a whole tone. 
 
 These rules for the recognition of intervals make use of a variety 
 of aspects : (a) symphony octave and fifth ; (6) paraphonic excellence 
 III, 3; (c) generalised 'harmonic' (i.e. polyphonic or paraphonic) dis- 
 position 7, diminished 5, augmented 4, II, and 2 ; (d) tendency towards 
 proximal interval of highly symphonic or paraphonic nature VII to 
 octave, VI and, still more, 6 to fifth, 4 to major third. Other distributions 
 of these aspects seem to be possible, for example : (a) symphony in its 
 grade 0, 5, 4; (6) paraphonic excellence or repose III (small and 
 bright), 3 (small and sad), VI (large and bright), 6 (large and sad, or as 
 discord tending to fall to fifth); (c) dissonance and size smallest 2, 
 next II, largest VII, next 7, middle size the augmented 4 and diminished 
 5; (d) proximity to a marked consonance VII to octave, 2 to unison; 
 second degree of proximity 7 and II; (e) generalised 'harmonic' 
 tendency as familiar. Or other rules might be formulated according 
 to the taste and fancy of the recogniser. 
 
 122
 
 180 THE FACTORS THAT MODIFY PARAPHONY [CH. 
 
 Similarly Bridge and Sawyer's rules for the recognition of single 
 tones, which presuppose a given (and retained) tonic, are : tonic (and 
 octave) at rest; leading tone up; submediant to the dominant; 
 dominant bright, bare, satisfying to the ear; subdominant dull, to 
 the mediant; mediant a calm, peaceful sound, on which the ear may 
 rest; supertonic to the tonic. 
 
 We do not feel any surprise at the major seventh's reminding us 
 of the octave nor of the minor second's suggestion of unison. In each 
 case the other interval is nearly there, as far as the volumic proportions 
 are concerned, and is a very characteristic thing. Apart from the 
 tendency of the fourth to the major third, what is most notable is the 
 connexion established between the major sixth and, in a stronger 
 degree, between the minor sixth, and the fifth. We might have expected 
 both these intervals to share the repose characteristic of the thirds. 
 The major sixth undoubtedly does so markedly; and it is debatable 
 whether it does not really do so much more than it points to the fifth. 
 But in the minor sixth this latter tendency is indisputable. Here, then, 
 we have another example to the fourth a paraphonic-consonant 
 interval with a tendency of 'resolution.' 
 
 In fact, we have more an exaggeration of the feature of dissonance 
 often ascribed to the fourth : for the minor sixth at least in equal 
 temperament is an undoubted dissonance whenever it functions as 
 an augmented fifth. It is so even when the chord it appears in otherwise 
 contains no dissonance at all ceg$ (two major thirds). When we 
 substitute e\> or/ for e, we get (ce\?a\>) the first inversion of the common 
 chord on the major tonic a [7 (= g$ in equal temperament) or the* second 
 inversion of the minor common chord on/ (cfa\>). Both of these chords 
 are generally consonant, apart from the bass fourth in the latter. In 
 particular the minor sixth is now consonant, although in the six-four 
 chord it still shows a tendency to fall to the g (the 'dominant,' as it 
 were, of itself and of the rules quoted above). 
 
 Of course the problem of the dissonance of c(e)g% is not solved by 
 a simple reference to the beating of partials. For although the third 
 partial of c (= g 1 ) and the second partial of g$ (= g 1 ^) beat with one 
 another upon a c of 144 vibrations per second 18 times a s'econd, 
 if these partials occur in the given primary tones, we must not forget 
 that the fourth partial of c (= c 2 ) and the third partial oi e(= b 1 ) beat 
 36 times a second, or, if the interval is lowered an octave, 18 times a 
 second. If the major third is not a dissonance on 144 vibrations, it 
 cannot be a consonance on 72 vbs. if cg$ is a dissonance on 144 vibrations
 
 xx] THE FACTORS THAT MODIFY PARAPHONY 181 
 
 because of its 18 beats per second. The same argument applies to the 
 interval ca\>, which would give 27 beats a second. We must, therefore, 
 look for some other difference than that of a few beats per second. 
 After all, the dissonance of a major second has only some 25 beats 
 (between the primary tones) in this region, so that it should sound 
 better than the minor sixth just mentioned, unless we admit that the 
 beats of partials are of less effect than those of primaries. But why 
 should they be so? And if they are, and if cg$ has no partials, or at 
 least not those that make 18 beats per second (only one of the beating 
 partials need be absent!), then cg$ would be consonant. But we cannot 
 revive the problem of partials as a whole again here. 
 
 In both the chords ce^a^ and c/a|? we have a grouping of a minor 
 third, a fourth, and a minor sixth. They differ only in their distribution. 
 In ceg$ we have two major thirds and (in equal temperament) a minor 
 sixth. If we argue that in the latter chord the outer interval is heard 
 as an augmented fifth, we must ask : why? It is certainly often claimed 
 that the ear has the wonderful faculty of hearing what a chord should 
 be ideally instead of what it really is. Equal temperament is commonly 
 said to rest upon this basis. But how does the ear know what should be? 
 Why does the displacement of the middle tone by a semitone upwards 
 or downwards incline it to hear cg$ (= a}?) rather as a minor sixth and 
 consonant than as an augmented fifth and dissonant? Surely not the 
 pitch of the middle tone, but the intervals it forms with the outer 
 tones. Then a minor third and a fourth somehow call for a minor 
 sixth, while two major thirds call for an augmented fifth. But why 
 should not two major thirds in the first instance have called for cg% 
 to be a consonance as well as ca|j? If answer be made that g% is not a 
 tone of the original scale, why, we must ask, did not this relation of it 
 to two major thirds call for its introduction into the scale, as other 
 chordal connexions have called for changes in scales (e.g. the sharp 
 leading tone in the minor scale)? 
 
 The question is certainly difficult. It would be foolish to try to make 
 it look easy. Probably the best reason is the one that appears first in 
 the musical consciousness : that ceg%, because of its major third in the 
 bass and the approximation of the two other intervals to the corre- 
 sponding ones of the major triad reminds us so strongly of this triad 
 that we feel the discrepancy as a distinct jar. confusion, or dissonance, 
 just as we may feel a certain approximation to the portrayal of a familiar 
 face as a caricature or as an artistic offence. This feeling can only be 
 supported by the fact that the interval in question even as a minor
 
 182 THE FACTORS THAT MODIFY PARAPHONY [CH. 
 
 sixth stands very low in the scale of fusions. When the minor sixth 
 stands alone, mere proximity to the fifth would make it suggest the 
 latter very strongly and so appear dissonant. Its paraphonic capacity 
 would not yet have been exposed by its subdivision into a minor third 
 and a fourth. So we find a prevailing tendency in early music to class 
 the minor sixth as a dissonance (cf. 60, 2). 
 
 Thus we come upon the general problem of the natural tendencies 
 of intervals to suggest one another. Of this the problem of the resolution 
 of dissonances forms a sub-division. 
 
 Probably the chief consideration in the latter problem must be the 
 cogency of the melodic movement of the voices 1 that constitute the 
 dissonant interval an interval that impels melodic movement forwards 
 and so calls for 'resolution.' Thus, for example, the tritone is commonly 
 said to resolve generally as a diminished fifth inwards to a major third 
 and as an augmented fourth outwards to a minor sixth. If, on the 
 basis of this generalisation, we were to claim, or to attempt to establish 
 by experimental means, some special connexion between these intervals, 
 we should not only find ourselves greatly embarrassed by the existence 
 of other forms of resolution of the tritone, but we should find it hard 
 to justify the distinction between the two forms of the interval 
 diminished fifth and augmented fourth, and their so different resolu- 
 tions. A much easier and more natural solution would be to claim that 
 the tritone must generally 'resolve' by the most cogent motion of 
 the voices it carries on ; and, apart from other modifying circumstances, 
 the smallest step is the most cogent. Hence the best resolution is by the 
 movement of a semitone in each voice (cf. above, p. 172, note). And, unless 
 we are to leave the key of the chord to be resolved, this procedure would 
 give the commonest resolutions of the two forms of the tritone mentioned. 
 We should by no means thereby be precluded from making other 
 resolutions of a chord containing the tritone, such as by movement 
 of only the lower voice of the tritone by a semitone, or by different 
 movements of the other elements of the chord than those of the tritone. 
 We should expect the 'harmonic' value of all possible resolutions, 
 i.e. their frequency in the great masters or their beauty, to vary directly 
 with the melodic cogency of the different voices of the chord, not 
 
 1 Cf. D. F. Tovey's interesting remark: "Even the modem researches of Helmholtz 
 fail to represent classical and modern harmony, in so far as the phenomena of beats are 
 quite independent of the contrapuntal nature of concord and discord, which depends upon 
 the melodic intelligibility of the motion of the parts" (74).
 
 xx] 
 
 THE FACTORS THAT MODIFY PARAPHONY 
 
 183 
 
 forgetting, of course, that the cogency of a movement, though primarily 
 determined by its shortness, may be increased by other factors, such 
 as consonance (step of fourth, fifth, or octave), tonality (as in the leading 
 note), an established figure or 'sequence,' the symphonic coherence of 
 one or even two otherwise undetermined movements with the specially 
 motivated and cogent movements of the other voices, etc. 
 
 The range of ordinary variations may be exemplified more fully 
 with the interval of the minor seventh on the dominant in the keys of 
 c major and minor. It is obviously not here a case of the mutual 
 suggestiveness of intervals, but of the melodic movements of voices 
 resulting in different intervals which are then accepted in so far as 
 they are consonant or paraphonic or conform with the intended tonality 
 or are possible companions for the tones resulting from the other voices, 
 and so on. 
 
 ^^^ 
 
 
 
 
 
 
 
 
 
 
 
 -*- 
 
 
 Nor do we need to assume the existence of a mutual suggestiveness 
 between whole chords as such (or between 'harmonies'). Any such 
 connexions or suggestions that may occur and be felt are merely a 
 scheme or shorthand of melodic connexions, having as a scheme no 
 binding force or compulsoriness at all. That belongs essentially to the 
 melodic transitions in detail and in their mutual compatibility, as 
 already indicated. Thus in the generalised connexion between the 
 chord of the dominant seventh and that of the tonic there is clearly 
 a parallelism of the patterns that result from the combined melodic 
 transitions or resolution. Neighbouring tones of each chord lifted up 
 an octave retain their melodic connexions unchanged. 
 
 sl 
 
 In the case of the connexion of chords by tonality, we reach a new 
 level of melodic grammar, so to speak. Instead of constructing from 
 single words we make use of familiar phrases. The connexion between 
 dominant or subdominant and tonic or vice versa is so familiar and
 
 184 THE FACTORS THAT MODIFY PARAPHONY [OH. xx 
 
 their chords so characteristic that we come to know exactly what to 
 expect. Thus the melodic connexion establishes itself more easily. 
 Besides, and perhaps this is the root of the matter, in these progres- 
 sions the very symphonic effect produced may be just what is desired 
 at that point of the music, namely, a tendency towards arrest or close. 
 The existence of an effect due to the connexion of chords a third 
 apart has been claimed by Shinn, who believes in a grading of effect 
 in three steps, corresponding to fifth-fourth, sixth-third and seventh- 
 second intervals between the chords. To some extent Shinn fails to 
 distinguish these connexions from the intervals between the pairs of 
 voices in question. The establishment of this grading would require 
 an extended statistical foundation, chosen with due regard to the 
 elimination of other differences. The problem raised by Shinn is of 
 considerable importance, but the factual and deductive grounds for 
 his generalisation seem to be at present insufficient. 
 
 It has been claimed occasionally that downward motion favours 
 melodic continuity, and so tends to annul unparaphonic effects. Here 
 again we lack a proper basis from which to confirm or to oppose this 
 suggestion. We know that descending intervals are found harder to 
 judge at first than are ascending ones. Whether the intervals of melodies 
 descend oftener than they ascend has yet to be determined (cf. 16, 173 ff.). 
 The effect of a progression of roots of chords by leap of a third is said 
 to be generally more satisfactory downwards than upwards, especially 
 if two or three such leaps occur in succession (52, 56; cf. 60, 35).
 
 CHAPTER XXI 
 
 RETROSPECT AND THE OUTLOOK FOR THEORY 
 
 THE division of intervals into the three classes of symphony, paraphony, 
 and diaphony, which we have attained and expounded in the previous 
 chapters, stands in opposition to the progressive grading of intervals 
 derived from the studies of Helmholtz and Stumpf. In these the octave 
 forms an extreme fusion of apparent unity in simultaneity or smoothness 
 in transition between tones, from which there is a gradual variation 
 towards complete two-ness or harsh beating in simultaneity or complete 
 difference in succession. We have reflected briefly from time to time 
 upon this change of theory, but it seems necessary to consider it more 
 explicitly now. 
 
 Helmholtz's quantitative estimation of grades of consonance between 
 simultaneous tones was based upon mathematical calculations regarding 
 the roughness of beating between their component partials (20, 192 ff.). 
 In so far as a connexion between successive tones might be established 
 by their common partials a certain smoothness of transition could be 
 achieved without which tones would be simply different; but there 
 could be no harshness or dissonance. In the case of simultaneous tones 
 beating would also appear so far as the partials presupposed actually 
 existed in the tones; and the roughness felt in any interval would 
 vary with the partials contained in its tones. But in so far as the 
 partials might lapse without alteration of the grading of intervals 
 according to consonance and dissonance, the theory would be fatally 
 discredited. No force of memory could save it. It is necessary to recall 
 these points from a previous chapter because they compel us to face 
 the task of constructing a theory of consonance and dissonance without 
 the help of coincident or beating partials, except it may be as a secondary 
 or supporting factor making for confusion or loss of distinction. 
 
 Stumpf's grading is based upon direct experimental observations 
 confirmed and amplified by the work of others. His description of fusion, 
 especially in its highest grades, as approximation to the unity of a 
 single tone, is a true representation of the experimental judgments 
 and corresponds fully with the descriptive definition of 'symphony' 
 given by the Greeks, which is thus seen to be correct both as far as it 
 goes and inasmuch as it does not venture farther. We have adopted
 
 186 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 it in our exposition above, and have found a sufficient and satisfactory 
 parallel to it in the volumic coincidence of tones, at least for the cases 
 ef the octave and the fifth. These yield a volumic mass that in two 
 distinct grades resembles the symmetrical balanced mass of sound 
 that constitutes the pure tone. There is no doubt that this part of the 
 theory must be maintained in all future theories of consonance and 
 dissonance. It fully explains the symphony of the octave and the fifth. 
 But in the other two divisions of paraphonic and diaphonic intervals 
 our scheme fails to coincide with Stumpf's. It is only in so far as we 
 can look upon intervals as mere static and isolated masses of sound 
 that we can attempt to force the aspect of 'approximation to unity' 
 down the whole series from octave to major seventh. But this static 
 aspect is practically excluded from music and is hard to realise experi- 
 mentally unless with unmusical persons or with neglect of analysis 
 (cf. Kemp, 29) 1 . The new description is one that establishes a plain 
 continuity between intervals (music in two voices) and music in any 
 number of parts. In paraphony, we have shown, there is an equal 
 distinguishability of the tones of an interval, so that two melodies may 
 run side by side through it without any mutual interference. In diaphony 
 this confusion re-appears, although the tones are not now apparently 
 one, but, as we can readily infer from their harsh coincidence, obviously 
 more-than-one-like. The major seventh and the minor second strongly 
 declare their dissonance. But they do not necessarily limit themselves 
 to a semblance of two-ness, as the erroneous judgments of untrained 
 and unmusical ears often indicate (64, 37iff.,e passim). We have to 
 find a sufficient basis for this positive diaphony (or dissonance, which 
 in music is always felt as a positive harshness not sufficiently described 
 by mere two-ness or difference 2 ) in the stuff of tones, and to emphasise 
 it again without the help of beating partials. Even beating difference- 
 tones must be kept in reserve for any special purpose they may properly 
 be able to fulfil 3 . 
 
 1 Kemp and Kiilpe go much farther than Stumpf, who refuses to see any change 
 in a fusion owing to the presence of other tones. 
 
 2 A much better description is, for example, this: " discordantia est duorum sonorum 
 sibimet permixtorum dura collisio" (Quisdam Aristotdes, 6, 250). 
 
 3 A claim to the recognition of neutral grades of sonance was made by F. Krueger 
 in his lengthy discussion of Stumpf's criticism of his difference-tone theory of consonance 
 and dissonance. Krueger pointed out that the psychological relativity of the notions 
 of consonance and dissonance and the historical mobility of the border between them 
 was noted by Helmholtz. We have already seen how little real truth there is in this 
 historical appeal. Stumpf was also tempted to form a class of neutral intervals, the 
 sevens in particular, but was clearly aware that his notion of fusion could give no such
 
 xxi] RETROSPECT AND THE OUTLOOK FOR THEORY 187 
 
 Now the whole course of our new theory of tones advises us to look 
 around the main points of our actual conclusions for lines of develop- 
 ment rather than away from them to outlying regions or to adventitious 
 phenomena. Our grading gives three regions with special points in 
 each, namely : 
 
 Symphony. Paraphony. Diaphony. 
 
 Loss of distinction in Ease and equality Loss of distinction in 
 
 unitariness of distinction confusion 
 
 p, o, 5, 4 III, 3, VI, 6 T, 7, II, 2, VII 
 
 (We have added the prime to the head of the series as our study of its 
 functions in music has justified our doing so. The prime, as the older 
 writers on music so often said 1 , and as the modern (acoustical) theory 
 of music has always failed to understand or to explain 2 , is the greatest 
 consonance of all.) 
 
 opposition as is implicit in the concepts of consonance and dissonance. Krueger's theory 
 does not really justify his formation of a neutral class either, however true it may be that 
 various intervals might well be placed in either class of consonance or dissonance. And it 
 is difficult to see how such an appeal to a neutral class can save a theory from special 
 criticism after it has been shown that such very different theorists as Helmholtz and Stumpf 
 also noted the existence of intervals of doubtful or indifferent character and yet equally 
 failed to justify the formation of a neutral class. Krueger believed that an interval tended 
 towards neutrality of sonance in so far as it was sounded more gently or briefly, or was 
 raised towards the higher end of the musical range of pitch, or was extended by one or 
 more octaves, etc. (v. 31, passim). Neutrality is given when "for any reasons the con- 
 sonantal or dissonantal characteristics postulated by the theory are lacking or reach a 
 certain limit of clearness or of effectiveness in the whole sound, or, again, when one and 
 the same chord contains both consonantal and dissonantal characteristics, and these 
 two sets, not greatly marked, annul one another in the total impression" (32, SM). 
 Consonance is due to the coincidence of difference-tones, of which Krueger asserts the 
 existence of five or six (theoretically) for each interval; dissonance to the 'neighbourly 
 interference ' of two or more of these five or six, i.e. to their beating and indistinguishability. 
 etc. Thus we get an approximation towards unity and indistinguishability both in the 
 greater dissonances and in the greater consonances. Why two primary tones and two 
 Joud difference-tones should suggest more oneness than these two primaries with five 
 difference-tones that are proportionately weaker is not explained: especially when we 
 recall that the difference-tones need not, and in the vast majority of cases are not dis- 
 tinguished at all. Surely the fewer more distinct elements should be 'rougher' than the 
 fainter ones. Besides, Stumpf s criticism of the facts regarding these five or six difference- 
 tones robs the theory of its basis (cf. 68, 69 and 70). 
 
 1 E.g. Johannes de Garlandia (6, 104): "Concordantia dicitur esse quando due voces 
 junguntur in eodem tempore, if a quod una potest compati cum alia, secundum audit mn ... 
 Perfecta dicitur quando due voces junguntur in eodem tempore, ita quod una, secundum 
 auditum, non percipitur ab alia propter concordantiam, et dicitur equisonantiam, ut in 
 unisono et diapason." Unison, as already said, has only meaning for polyphony; for 
 only then can one sound be held and heard to be the conjunction of two. 
 
 1 Cf. 64, n: "the hypothetical fusion of the prime."
 
 188 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 In order that we may look around these intervals for some means 
 of progress, let us arrange them according to their place in the ordinal 
 series of an octave. Then we get : 
 
 Symph. Diaph. Paraph. Diaph. Symph. Diaph. 
 
 p 2, II 3, HI ? 4 T 
 
 o VII, 7 VI, 6 6 5 T 
 
 The two ends of the series are constituted by the greatest symphonies; 
 next them stand the greatest diaphonies ; a lesser diaphony leads through 
 the two grades of paraphony to a middle region in which stand the 
 consonances of the fourth and fifth, separated by the diaphonic tritone. 
 This arrangement suggests a regular or periodic change from the extremes 
 towards the middle region, in which range the diatonic intervals mark 
 out familiar points. 
 
 But, we must ask, do these intervals really designate points in the 
 range of intervals included within the octave ? If both the minor and 
 the major second are diaphonies, and both the minor and the major 
 third are paraphonies, should we not rather speak in general of a 
 diaphonic or paraphonic region? And should we not also in particular 
 speak of a region having, for example, the paraphonic degree of the 
 minor third? (Remember again that we have lost the treacherous aid 
 of beating partials !) 
 
 Arguments of some strength may be urged in support of such a 
 view, startling and even outrageous as it may at the first glance seem 
 to be 1 . These rest upon facts that are fully familiar to students of 
 
 1 One can see, however, that Stumpf felt himself drawn towards this conclusion 
 (cf. 64, ITS fl.): "Doubts about the sinking of the curve between [major and minor thirds, 
 etc.] have another special cause in that these intervals differ only by a semitone and so 
 the intervening are always apprehended as a deviation from them, as a mistuned third 
 etc." So, also, Helmholtz, who wrote (15, 200): "two simple tones making various intervals 
 adjacent to the major third and sounded together will produce a uniform uninterrupted 
 mass of sound, without any break in their harmoniousness, provided they do not approach 
 a Second too closely on the one hand or a Fourth on the other. My own experiments 
 with stopped organ pipes justify me in asserting that however much this conclusion is 
 opposed to musical dogmas, it is borne out by the fact, provided that really simple tones 
 are used for the purpose." The later experimental work of Stumpf and Meyer (60) has 
 disproved this in so far as the ability to recognise the interval with great approximation 
 to its pure form is concerned. (Helmholtz went so far as to say it was impossible to tune 
 perfect major or minor thirds on stopped organ pipes or tuning-forks without the aid of 
 other intervals.) Stumpf and Meyer's results prove the ability to find the almost exact 
 interval even with successive tones. Here there is presumably no 'fusion' of any kind, 
 beats, difference-tones, or 'fusion' proper, so that we are all the more thrown back 
 upon an accurate sense of interval (to which we have given a positive basis in our theory 
 in volumic proportions, and) which is presumably most developed in the most highly
 
 xxi] KETROSPECT AND THE OUTLOOK FOR THEORY 189 
 
 hearing and of music, but that have been rather coldly entertained by 
 those who have aspired to construct a theory of the foundations of 
 music. This neglect is, of course, natural and inevitable so long as the 
 facts in question seem to be not only incompatible with the prevalent 
 theoretical point of view, but even to be artificial and arbitrary dis- 
 tinctions or classifications unattainable from the straight roads of 
 theory. 
 
 One argument is familiar to all those who possess absolute ear and to 
 others from their records. Such a person does not, for example, hear the 
 progressive transition from a good d to a good e as a sharp lapse of 'd' 
 into ' something not d, but higher than d,' then ' d%,' then ' flatter than e? 
 and finally 'e,' or the like; but as a region of 'd colour' surrounding the 
 ' best d,' bordering on a region of d% including a smaller optimal region, 
 passing into a new region of e leading to a best e, and so on. Each 
 tone name characterises not one rate of vibration or a very narrow 
 zone defined by the range of hardly distinguishable differences, but a 
 relatively large zone of perfectly obvious differences. An absolute ear 
 also adjusts itself readily enough to differences in the pitch to w^hich 
 a musical instrument, such as the piano, may be tuned, if the difference 
 is not great enough to carry a tone into the zone of its chromatic neigh- 
 bour. In this case keys are interchanged, c major becoming d\> major 
 or b major. In some cases of very rigid absolute ear, however, even 
 smallish differences of tuning are disturbing. 
 
 Another is the similar usage so characteristic of music of applying 
 the same pitch names to zones of pitch. Thus, e.g. we have not only 
 dfy, d and d% but also dfyfy and dx (ftft) 1 , terms which are acoustically 
 so anomalous, as they are on instruments of equal temperament exactly, 
 and in any case very nearly, equal to the pitches of c and e. Similarly 
 the names of the intervals are not only the same for their major and 
 minor forms, but they are extended to include augmented and diminished 
 forms. An augmented second, for example, is more or less the same as 
 a minor third as far as pitch is concerned; but, as music has always 
 claimed, the augmented second partakes of the diaphony of the major 
 second, not of the paraphony of the minor third. And an augmented 
 
 trained observers such as Stumpf and Meyer had. Hence in observers such as Helmholtz, 
 who do not possess this faculty, we may assume we get an approach to the effect of the 
 consonantal value of the tones apart from their intervallic precision and apart from 
 difference-tone values which Helmholtz obviously failed to make use of in these cases. 
 Hence a neutral (or as I call it, a paraphonic) range from second to fourth or thereby. 
 
 1 The original points for these are, of course, not rf, but (ty and <IZ respectively, so that 
 there is a second grade of flattening or sharpening here only in a terminological sense.
 
 190 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 or diminished interval, when it occurs in full force, as between a (7 and 
 6 in a minor scale is found to be peculiarly hard to sing. When the 
 same transition of pitch occurs in a true minor third, as from c to efy 
 in c minor, no such difficulty is felt. 
 
 We may recall a quotation from Stumpf (p. 144, above) : " That one 
 and the same unmodified pair of tones should now fuse more and now 
 less according as we apprehend it as c-efy or as c ~d% i g ut of * ne question, 
 because fusion is a function of the two sensations or of their physio- 
 logical bases and can change only with these same" (71,328). This 
 utterance implies the very sharp distinctions of the points of the scale 
 and the fusions peculiar to them that we have already mentioned. 
 But, if pitches and degrees of paraphony belong, not to points optimally 
 and only, but also in some degree to zones or regions, there seems to 
 be no insuperable barrier to our extending these zones for various 
 reasons and to ascribing to a tone the pitch name, or to an interval 
 the paraphony, of the zone within which it is included. Then in con- 
 nexion with these musical reasons of key relationship, modulation, etc., 
 we shall apprehend a tone of a certain number of vibrations as a modified 
 d, i.e. as d$, not as efy, with which it may more or less coincide, and we 
 shall then attach to it the pitch name of the (extended) region we ascribe 
 it to and the paraphony of that region, namely the diaphony of the 
 seconds. But its diaphony will not be greater than that of the major 
 second as is the minor second's, but less, of course; still, however, 
 diaphony, and not the paraphony of the third. The paraphony of an 
 interval will be determined primarily and chiefly by its precise nature 
 as an interval, but within any zone much will also depend upon the 
 circumstances of our approach to it in the flow of the music. 
 
 Reverting to our series within the octave we may now note further 
 that the great symphonies are bounded by the strongest diaphonies. 
 In the former there is a special fit or balance of the tones; in the latter 
 we have the worst coincidences of volume, as soon as the range of hardly 
 perceptible difference is passed. The zone of symphony in the octave 
 and fifth is very narrow, allowing of no degrees, no major or minor 
 forms. In the diaphonies bordering on the symphonies the pitch centres 
 of the tones either encroach upon one another (small intervals) and so 
 make each other hardly distinguishable, or the lower end of the volume 
 of the upper tone and the pitch centre of the lower fall close together 
 (large intervals), making a notable blemish in the centre of the lower 
 tone, and, in the major seventh, an approximation to the octave which 
 it thus strongly suggests. We probably do not shift our point of attention
 
 xxi] RETROSPECT AND THE OUTLOOK FOR THEORY 191 
 
 in these two groups from one side to another of the pitch of the lower 
 tone. We hear each tone in a mass as a whole, including its pitch and 
 the length of its volume; then its paraphonic nature will be decided 
 by whatever excessive proximity of parts or relative freedom and 
 independence of them there is on the whole. 
 
 In the paraphonies we reach a zone of volumic proportions in which 
 the pitches of the tones stand well apart, are therefore easily discriminable 
 without mutual interference, and in series can readily form distinct 
 melodic lines, not running together or blurring one another. 
 
 Round the fifth again we find a border of diaphony in the tritone 
 and in the minor sixth when it appears as an augmented fifth or when 
 it is, as in early music, not marked off by its place in chords as a para- 
 phonic sixth. This fluidity of character confirms the extensibility of 
 zones we have already noted. And it also seems a sufficient explanation 
 of the fact that in the sixths the lesser paraphony follows upon the 
 greater, not conversely as in the thirds. If the fifth stood not where 
 we find it, the interval next beyond the major sixth (i.e. the minor 
 sixth) would presumably be a better paraphony than it actually is. 
 
 The only point that remains for discussion concerns the fourth, 
 that old bone of contention. Our scheme tempts us anew to consider 
 it a dissonance, less diaphonic than is the tritone. So one might plausibly 
 account for its having been so often termed a dissonance. But even so 
 it might at the same time be held, like the sixth, to be includible in the 
 paraphonic zone of the thirds, being then a little more paraphonic 
 than the major third. But, on the other hand, we cannot get round 
 the fact that the fourth is a consonance of third, though low, degree. 
 To what extent its being the inversion of the fifth is responsible for this, 
 it might be difficult to decide. The connexion felt between its symphony 
 and its position in the upper voices of chords would support any doubt 
 as to its being in itself symphonic, as would also the large (statistical) 
 gap between the fifth and the fourth and the small gap between the 
 latter and the thirds (cf. 77, 58). Our failure to find a form of distinctive 
 volumic balance for the fourth would tend towards the same conclusion. 
 So we should infer that the fourth is a paraphony or even a diaphony 
 whose cousinship to the fifth has given it the rank of symphony. This 
 conglomeration of relations would suit the unstable character of the 
 fourth in music. An incautious fourth from the bass would then be a 
 sort of augmented third or a diaphony of lesser degree than the tritone. 
 It would have the fluid nature of the minor sixth between diaphony 
 and paraphony, sometimes 'resolving' therefore into an ordinary (major)
 
 192 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 third. There is in the relation of inversion as such no reason for any 
 transference of symphony from the fifth to the fourth. But when the 
 fourth occurs as the top of a major common chord, it is then the comple- 
 ment to an actual fifth. 
 
 But if the range of the octave is characterised by these wide regions 
 of paraphony and diaphony, what, the reader may ask, accounts for 
 the very precise form of our present diatonic scale and for the perfection 
 that so many claim for just temperament. Here we return to the problem 
 of the scale, which is primarily the problem of the skeleton of divisions 
 of the octave that will at any time offer the greatest and most convenient 
 scheme of intervals and chords, etc. The octave and fifth are inevitable 
 land-marks of great precision and force. The fifth may be taken upwards 
 or downwards, yielding cfgc 1 . And from the whole tone thus derived 
 we may get various farther divisions of which the chief type attempts 
 to subdivide the two large intervals the two 'tetrachords.' It is a 
 familiar fact that many forms of subdivision have been tried and 
 tolerated as more or less permanent modes. We ourselves have two the 
 major and minor, the latter of which the theories that have worked 
 with harmonic partials have never been able to justify. And in bagpipe 
 music we find a neutral third midway between our major and minor 
 sections. The extensibility of zones, it may be noted in passing, is 
 richly confirmed in the familiar tendency of the ear trained upon our 
 two modes to hear the intervals of exotic music as approximations to 
 the intervals from the same zone that occur in our music. 
 
 Of course the burning question is : why has our harmonic style so 
 favoured our two sections in the precise form they have taken, and 
 especially the major one? Here the appeal to the correlation with the 
 ratios of harmonies seems inevitable. To deny may seem like wilful 
 scorn of a providence governing musical theory. But at this point 
 difference-tones seem to be much more important than upper partials. 
 For the latter are not only accidental and distant, but as Macfarren 
 said, they should really create quite a babel of confusion; whereas the 
 loudest difference tones of 4, 5, and 6 would necessarily be 1 (twice), 
 2 (twice), 3, and 4; or with the primaries 1 (loud), 2 (loud), 3 (not 
 weak), 4, 5, and 6. Here we have two octaves 1-2 and 2-4, and two 
 fifths 2-3 and 4-6. A deviation from 5 in either direction would upset 
 the two octaves and one of the fifths, thus creating in the whole sound 
 a somewhat indefinitely located, but quite noticeable, harshness. The 
 minor third, on the other hand, is sufficiently justified by its being the 
 remainder of the major third from the fifth (which two would between
 
 xxi] RETROSPECT AND THE OUTLOOK FOR THEORY 193 
 
 them yield all the tones of the major diatonic scale). Any scale deter- 
 mined by these two intervals would also yield minor third chords, 
 which would thus become familiar parts of the system, tolerated a& 
 such, in spite of their less symphonic resultant, which is familiar to every 
 musician. 
 
 Two points may be emphasised with reference to the above. First, 
 we are not here attempting to justify the thirds ab origine by this means; 
 they carry their own justification as paraphonies within them; the only 
 question is : why has our harmonic music so favoured the particular 
 points of the paraphonic zone at which the major and minor thirds 
 stand? Second, we do not forget that the major mode was by no means 
 the fount and origin of all music; it is rather the culminating result 
 of a long development. But it stands in modern music as the logically 
 prior skeleton of our system, and as such it requires special justification. 
 
 It is open to the musician, of course, to make many other sections 
 of the octave than our two, and to build a perfectly coherent music 
 of somewhat limited scope upon them. This may be forcibly extended 
 even some way into the range of polyphony. But the fascination of 
 the great consonances gradually drags it into the lines of the two chief 
 steps of paraphony and the great systems they yield. 
 
 With this free outlook we seem able to do justice to all kinds of 
 music without the ridiculous restrictions and sophistries of the harmonic 
 theories. Of course, all the theories try to bring themselves into line 
 with music so as to explain its ways; and in so far as theories succeed 
 in doing so, we may be all taken as in agreement with one another. 
 The only question is : whose basis of explanation is the primary one ? 
 And we may claim that honour for ours for two reasons : (1) the basis 
 of explanation is inherent in the primary tones concerned in the 
 phenomena to be explained; (2) the new facts we have built upon have 
 not been derived from sources external to music so as now to be 
 problematically read into music, but they have been derived from the 
 empirical generalisations of music itself. 
 
 The explanation suggested for the nature of the intervals between 
 the fifth and the octave would obviously hold equally for the intervals 
 greater than the octave up to the twelfth. For in the ninths the ends 
 of the upper volume will fall near the pitch centre of the lower tone 
 and will so mar its outline, while in the tenths and eleventh these 
 points will draw apart so that they will be readily distinguishable and 
 independent. In the twelfth we meet again (cf. 77, 68, inf.) with a 
 sort of balance or symmetry of volumic parts, the upper half of the 
 
 w. F. M. 13
 
 194 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 volume of the lower tone being divided into three equal parts by the 
 pitch and the lower end of the upper tone's volume. So we may well 
 have diaphonies in the region on either side of this point. But there 
 seems to be no further ground for establishing paraphonies, such as 
 the compound sixths. Perhaps the diaphonies around the double 
 octave might pass as blemishes on the division (4, 2, 1, 1) peculiar to 
 the latter. 
 
 At this point, feeling we had pushed the range of the direct and 
 inherent distinctions of intervals well out beyond the octave without 
 
 Relative diminution of volume in higher tone 
 
 _ 
 J-T 
 
 X 
 
 x-2 
 
 25 16 
 24 15 
 
 10 9_ 8 
 J 8 7 
 
 9 IS 
 
 5 8 
 
 . = Diaphonic 
 
 P Z. = Paraphonic Zone 
 
 Fig. 6. 
 
 6' = Symphony 
 
 appealing to any extraneous or secondary factors, we might perhaps 
 venture to ask the functions of memory to support us or invite the 
 aid of harmonics. These functions must certainly accrue at some time or 
 other; for there can be no doubt that we come in time to be thoroughly 
 familiar with proportions and divisions of tonal volume by mere memory. 
 As we have seen, all our sense of interval is founded upon that ability. 
 But we must take all care to justify our musical distinctions by powers 
 that dwell in the stuff of tones as they are actually presented to us,
 
 xxi] RETROSPECT AND THE OUTLOOK FOR THEORY 195 
 
 before we appeal to any outlying circumstances or to habit. We must 
 bring our engine into motion with its own steam, ere we can expect 
 to steady its action by its acquired momentum. 
 
 The results of the above suggestions may be embodied in graphic 
 form (fig. 6). The height of the curve represents in some fashion the 
 comparative diaphony or paraphony or symphony of the intervals 
 below it. The height has been taken in most cases as the denominator 
 of the fraction by which the volume of the higher tone is less than the 
 lower, thus -fa or thereby (for a quarter tone), ^, ^, , , \, , similar 
 values hold in the case of the intervals less than the octave from just 
 below the octave, through major and minor seventh to major sixth. 
 The intervals on either side of the fifth must, of course, have high 
 values, sinking in the tritone perhaps to the value of the minor seventh, 
 and in the minor sixth to a somewhat lesser quantity, yet still greater 
 than that of the major sixth. But these regions cannot yield the fierce 
 'collisions' that we get near the octave and the prime. The high value 
 of the augmented fifth lies near the minor sixth as the quick fall of the 
 curve in some way indicates. A zero value may be assumed to lie some- 
 where in the paraphonic zone 1 . The very small symphonic zones cannot 
 properly be joined to the rest of the curve by lines; for there is no gradual 
 transition from them to their neighbouring diaphonies through an 
 intermediate paraphony. So it would be useless to indicate their relative 
 degree of symphony by vertical position in the figure. There is a transi- 
 tion only where the lines are drawn out. As already said, the ranking 
 that should be given to the fourth is doubtful. The figure, of course, 
 takes no note of any differences of smoothness that may be due to 
 coincidence of partials or of difference-tones, as these are only subsidiary 
 differences that would be ineffective apart from the differences inherent 
 in the tones themselves that the figure attempts to indicate. But the 
 beating of upper partials and of difference-tones would aggravate the 
 original diaphony of the primaries most of all in the immediate neigh- 
 bourhood of the great consonances, as has already been indicated 2 . 
 
 1 According to Krueger neutral intervals are to be sought "in a middle region of 
 vibratory ratios close beyond the sevens intervals," i.e. in the first place in such ratios 
 as 6: 7, 4:9, 5:9, 7:9 (31, aw). 
 
 J Our figure agrees very well with the similar figures that may be constructed from 
 the experimental ranking of the intervals of the octave given by C. F. Malmbcrg (36, in) 
 for the 'factors' or aspects of intervals named by him Smoothness (relative freedom 
 from beats), Purity (resultant analogous to pure tone) and Blending (a seeming to belong 
 together, to agree. The curves for these three are in their general course very similar. 
 Malmberg's fourth factor of Fusion (a tendency to merge into a single tone, unanalysable) 
 
 132
 
 196 RETROSPECT AND THE OUTLOOK FOR THEORY [CH. 
 
 Thereby the extent of the zones of original symphonies has perhaps 
 been somewhat reduced; and the point of optimal symphony has been 
 greatly sharpened and emphasised by the smooth system of symphony 
 throughout difference-tones, primaries, and partials that then accrues. 
 In the case of intervals beyond the octave, and, still more so, beyond 
 the double-octave, the ordinary harmonic partials present in instru- 
 mental tones would carry forward the basis of original paraphony 
 which we have established for the first octave and the greater part of 
 the second. And the beating of these components would similarly 
 mark out the neighbourhood of these distant symphonies, and so on. 
 In considering the need to appeal at some time to the memory of 
 volumic proportions and their systems in music it must not be forgotten 
 that for the consonance of successive tones no help can rightly be 
 sought either from partials or from difference-tones. The relations 
 upon which we have founded our constructive theory are the only 
 ones that then remain over. Here the step from them to the work of 
 memory is therefore direct. And the memory of volumic proportions 
 is, as we know from vision and from the sense of time, a well established 
 and well trained faculty. 
 
 In connexion with successive tones we have already shown how the 
 two great consonances would make themselves felt even in music that 
 hardly knew such a thing as simultaneity of tones. The basis of these 
 successive consonances is exactly the same as that of the simultaneous 
 ones. And to the fifth the fourth is bound almost hand and foot, as a 
 distinctive grouping of tones. For the other intervals less than the 
 fifth the specially emmelic ones no clear definition is given in mere 
 succession. Their determination will therefore be decided by two chief 
 conditions : the melodic cogency of the step, which is the greater the 
 smaller the step (no doubt within a certain limit) ; and a basis of ease of 
 distinguishability between the tones of the step which, we may suppose, 
 is closely akin to the distinctiveness of pitches which produces paraphony 
 
 shows itself graphically as a pure distance- judgment, correlated to separateness of pitches. 
 Its course of steady decrease from minor second (maximum) to major seventh (minimum) 
 is broken only by a slight rise in the fifth and a rise to greatest maximum in the octave. 
 This indicates again the unitariness of these intervals. Probably Blending and Purity 
 are expressions for the same thing in Malmberg's tables. He also found that, "when by 
 the use of two sets of tuning-forks, the just intonation was compared with the tempered 
 intonation, no difference of ranking of the intervals large enough to affect the order 
 resulted from the difference in temperament" (p. 107). 
 
 For evidence of the way in which partials, etc., mark out points in the paraphonic 
 zones where pure tones give little if any differences, see 28, 491 1 .
 
 xxi] RETROSPECT AND THE OUTLOOK FOR THEORY 197 
 
 as against diaphony in simultaneous tones. It is in agreement with 
 this that we find in monophonic or homophonic music so much freedom 
 and variation in the subdivision of the two ' tetrachords ' left by the 
 fifth and fourth. But as soon as polyphony has revealed its great 
 preference for such thirds as we have adopted as fundamental, all the 
 melodic or successional aspects of music would necessarily come under 
 control. And we have already shown that a basis exists for the 
 (attitude of) apprehension of simultaneity in succession that we find 
 in our music and that is able to make a succession of tones rather a 
 harmonic whole than a melody, or to give to a melodic succession the 
 structural form and the atmosphere of a harmony. So strong in fact 
 is this colouring of harmony in many of our melodies that various 
 theorists have sought for a purely harmonic foundation for melodic 
 figuration, a procedure which is surely misleading. This harmonic 
 tinge in melodies only appears when the structure of the sequence is 
 such as to invite that sort of reminiscence from a mind already full of 
 harmonic experience. But it never appears even from such a mind 
 when the melody proceeds mainly by small steps. These cogent move- 
 ments are strong enough to repel any incoherent associations towards 
 dissonance that we might for once try to weave into them.
 
 CHAPTER XXII 
 
 SYNOPSIS OR OUTLINES OF INSTRUCTION 
 
 THE following may be taken as a summary of what has been expounded 
 in the previous pages. It will also serve as a sort of introduction to the 
 positive results I have reached. There are many readers who will be 
 anxious to see what it all amounts to when the criticisms, evidences 
 and authorities have been laid aside. Besides, no time should be 
 lost in bringing to the succour of the student of music (and perhaps 
 of harmony in particular) what may be firm ground for his intellectual 
 efforts with that art. Until now he has been fed with unexplained 
 empiry or with obviously unreasonable messes of harmonics, both of 
 them diets of no sustenance for the hungry intellect. The teacher 
 of musical theory will perhaps be able to follow this brief sketch of 
 its foundations and to give his pupils a reasonable and continuous 
 introduction to the art. For that reason I have put the sketch into the 
 form of an outline of instructions, in which the topics are merely arranged 
 and briefly stated in their systematic order without being expanded 
 into detail. That extension the teacher will readily make for himself 
 according to his inclinations. 
 
 1. The difference between tones and noises is fundamental. Tones 
 are smooth, regular, balanced, symmetrical; noises are the contrary. 
 
 2. Ordinary musical tones are really blends of tones (fundamental 
 and other partials or harmonics). Analysis reduces them, by methods 
 that are already generally familiar, to a single series of pure tones. 
 Chap. i. 
 
 3. Tones differ in pitch, whereby they fall into an obvious order. 
 They differ also in volume : those at the one end of the series are large, 
 massive, voluminous; those at the other end are small and point like. 
 Between the two ends there is gradation of volume. Tonal volume 
 we feel to be a quantitative difference in tones. 
 
 4. Volumes commonly consist of parts. The question thus presents 
 itself : does a tone consist of parts? Have different tones any parts in 
 common? 
 
 Consider simultaneous tones first. Play c with d, e, f, g, a, b, c 1 , 
 d 1 , e 1 , ... (cd, ce, cf, etc.). Notice :
 
 CH. xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 199 
 
 (a) that the two tones never seem to get entirely away from one 
 another; c and d seem to confuse each other very much, they seem to 
 be mixed up with one another in confusion ; c and e seem to be less : in a 
 heap'; and so on progressively; c and d 1 are relatively free from one 
 another. 
 
 (6) that, nevertheless, the series of pitches of d, e, f, g, etc. is the 
 same in combination with c as it is when these tones are played singly 
 in a series; the series stands untouched in spite of the fact that the 
 tones of each pair never become quite free from one another. 
 
 5. Now write down the series of pitches as dots (or as points 
 separated by a short space, representing the fact that between c and d, 
 for example, other tones could be distinguished c#, d\>, quarter-tones 
 
 Fig. 7. Showing the relation between the pitch and the volume of the tonal series. 
 
 In this diagram the volume (length of line) of every higher tone must be supposed to be 
 projected upon the volume (line) of the lower tone. They have been separated into 
 an ascending series only for clearness of exposition. The dots represent the pitches 
 of tones. 
 
 or eighth-tones, etc.). Then put volumes round the dots to satisfy 
 paragraph 1 (tones are balanced, symmetrical) and paragraph 4a (no 
 two tone volumes are entirely clear of one another). It will be found 
 that the volume of every tone will have to go quite up to the right- 
 hand end of the pitch series where the dots for the highest tones lie. 
 Thus (cf. Chap, n) : Figure 7. 
 
 6. Repeat this series cd, ce, cf, etc., noting this time, instead of the 
 gradual change throughout the series, the features that distinguish 
 cd from ce, from eg, from ce 1 , etc. For this purpose it may be necessary
 
 200 SYNOPSIS OR OUTLINES OF INSTRUCTION [CH. 
 
 to compare the pairs of tones, not as they stand in a series, but in some 
 other order, pair against pair. In cc 1 and, to a noticeably less extent, 
 in eg we observe a balance or symmetry of the whole sound that reminds 
 us of the appearance of a single tone. 
 
 How can we represent this with the lines of figure 7 ? If the line for 
 the higher tone of the pair grows progressively smaller, as is there 
 indicated, we shall at a certain point get this figure (fig. 8) where 
 the higher tone's volume just fills the upper half of the lower tone's 
 volume. This arrangement would surely give the nearest possible 
 approximation to the simplicity and balance that characterise very 
 smooth (or pure) single tones. Hence we may ascribe it to the octave. 
 The volumes of tones an octave apart would, therefore, be to one another 
 in the proportion of 2 : 1. 
 
 Another form of balance would be given when the two new points 
 of the upper tone lie equally far on either side of the middle (pitch) 
 point of the lower tone, thus balancing each other around it. But the 
 whole would have more points to distinguish it from the single (pure) 
 
 Fig. 8. Showing the relations between the volumes (and pitch-points) of a tone and 
 
 its octave. 
 
 tone than the octave has. Hence it will not be so unitary in effect as 
 the octave. This arrangement (which figure 7 shows must arrive at 
 a certain place in the scheme) we may ascribe to the fifth. And the 
 volumes would be in the proportion of 3 to 2 to one another. 
 
 7. Hence we now know precisely how simultaneous tones overlap, 
 and we can correct (or annotate, as the case may be according to the 
 instructor's procedure) figure 7 to that effect. The volume of a tone 
 decreases by half for every octave, by 3 : 2 for every fifth. (These 
 proportions have actually been embodied in figure 7.) These proportions 
 happen to be the same as those that hold for octave and fifth between 
 the rates of the physical vibrations required for these intervals. For all 
 the other intervals the ratios of their volumes follow as a matter of 
 course (just as do the ratios of the rates of physical vibration). It is 
 the balance of volumes that characterises the octave and the fifth as 
 heard intervals in respect of their fusion; they form a sound that is 
 more like one tone than other intervals do : the octave most and the 
 fifth next. The other intervals do not differ so markedly from one 
 another in their appearance of balance as do the octave and fifth from
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 201 
 
 one another and from all the rest. But they have been graded by careful 
 experiment in the following way : fourth, major 3rd, minor 3rd, major 
 6th, minor 6th, tritone, major 2nd, minor 7th, minor 2nd, and major 
 7th. The volumes corresponding to these intervals seem likewise to 
 grow less and less balanced. Their characteristic points come more and 
 more into conflict with one another. But for a fuller account of these 
 grades we must await certain lines of information to be given below (18). 
 Chap. m. 
 
 8. The same relations can be established between successive tones 
 (in the matter of the relative positions of their volumes) as hold for 
 simultaneous tones. Only it is not now primarily a question of balance 
 or rivalry. Sequences of tones are not of themselves dissonances (or 
 consonances). But they can play the part of dissonances (or consonances, 
 as the case may be), in music (as arpeggio chords); and they then call 
 for the same treatment. This may be easily explained by supposing 
 that upon occasion we can take a point of view from which we notice 
 the balance of parts formed by two successive tones. These are 
 apprehended in projection against one another as parts of a whole. 
 In fact there is probably a special charm and value in these relations 
 of balance or unbalance, because they are not accompanied by the 
 actual fusion or clashing peculiar to simultaneous tones. Chap. v. 
 
 9. Interval is not the same thing as fusion (or as consonance and 
 dissonance). Thirds and sixths are similar in fusion, but very different 
 as intervals. Interval is the constant proportion between the sizes of 
 the volumes of any two tones of absolute pitch. The same proportions 
 can be got in an indefinite number of lower and of higher tones. Then 
 the two volumes are proportionately larger or smaller. Chap. vi. 
 
 10. The musical range of pitch (approximately A 2 to c 5 , or seven 
 octaves and a third) is the range of difference in physical vibratory 
 rate within which, in the volumes of the tones evoked, there is a constant 
 volumic proportion (or interval) for any given ratio of vibrations. At 
 the upper limit of the musical range the proportion begins to be too 
 large (e.g. 2:1 + instead of 2 : 1 exactly for the octave ratio of vibra- 
 tions), so that the highest tones appear to be somewhat flat; at the 
 lower end the proportion is too small (1 : 2 +) so that the lowest tones 
 appear not to be low enough, or as low as they should be according to 
 the ratios of their rates of physical vibration. (These restrictions seem 
 clearly to be due to the limits of size and of subdivision of the responsive 
 tissues of the cochlea). Chap. vn. 
 
 11. The predominance of the central point of the volume of a pure
 
 202 SYNOPSIS OR OUTLINES OF INSTRUCTION [CH. 
 
 tone and the smooth grading of intensities round that pitch-point set 
 us a pattern for the apprehension of tonal masses in general. Except 
 in so far as melodic interests draw us away, we tend to apprehend the 
 mass centrally, i.e. it seems to have the pitch of its lowest component. 
 When the melodic interests of different 'parts' or voices are equally 
 developed, the bass will be the most effective. For this reason, also, 
 we reckon intervals always upwards; intervals are more readily learnt 
 and recognised in ascending form; descending intervals have to be 
 learnt to a greater or less extent as something fresh; and there is a 
 similar disconnexion between successive and simultaneous intervals. 
 Chap. vin. 
 
 12. The harmonics or upper partials that constitute the blend of 
 musical tones are heard synthetically : i.e. we do not strive to concentrate 
 the attention upon them separately to the exclusion of the rest of the 
 blend ; and they are intentionally kept so weak that this will be difficult 
 to do. We hear them then merely as a pleasing change in the ' surface ' 
 of a tone's volume, departures from the ideal smoothness and graded 
 balance of pure tone that add interest to the whole while leaving its 
 balance and smoothness approximately intact. 
 
 Difference-tones (of which the chief are : h-l, and 2l-k) occur only 
 when two primaries are sounded together. Their presence is generally 
 overlooked, not only because they are heard synthetically in the whole 
 tone-mass, giving it a touch of lowness, but because they generally 
 arise only within the ear and so have nothing in the instrument corre- 
 sponding to them, and because they appear only when two (primary) 
 tones are sounded together. The separation of these primaries analytic- 
 ally is a task that exceeds the power of (perhaps) most people, so that 
 it is little wonder that the difference-tones pass unobserved. But the 
 chief difference-tones are not hard to hear. 
 
 The first and prevailing attitude adopted by a listener towards the 
 primary tones of a chord is likewise that of synthesis, which is founded 
 upon the inevitable overlapping of the volumes of tones (described 
 above). The ease of analysis that is possible for specially gifted ears 
 cannot, of course, annul this synthesis, although it makes the ear aware 
 of the component tones. Practice will tend towards the same analytic 
 ease, which naturally directly subserves the purposes of accurate and 
 beautiful musical production, etc. 
 
 Difficult directions of synthesis are favoured by different musical 
 methods, especially by melody ('horizontal' direction) and by harmony 
 ('perpendicular' direction). Polyphonic and harmonic music differs in
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 203 
 
 the prevalence in each of one of these methods as a basis of structure : 
 melody in polyphony, fusion in harmony. Chap. ix. 
 
 13. The connexions established between certain chords, whereby 
 they come to be recognised as inversions of a certain chord, are 
 founded upon the (volumic) pattern formed by the volumes of the 
 several over-lapping tones (fig. 9). This pattern is not obscured or 
 distorted by being continued upwards or downwards into the neighbour- 
 ing octaves. And the continuous pattern thus obtained contains all 
 the different inversions of the 'one' chord. 
 
 The octave is the only interval that makes possible such an extension 
 of pattern without any distortion of it. The fifth is quite unsuitable 
 for this purpose, high consonance though it is. This peculiarity of the 
 octave is referable to the special way in which its higher tone fits into 
 the upper half of its lower one. 
 
 '; ' ' -? f.... T ...^.....*.- All 
 
 V X Y Z 
 
 Fig. 9. The proportions of parts in the 'All' line are repeated from VX to XY to YZ in 
 
 reduced form (1 to \ to J). 
 
 But the connexion thus established between inversions does not in 
 any way annul their very important differences; it only gives them a 
 close connexion in spite of these differences. And this connexion is 
 not a merely formal one, based upon an algebra of transposition; it 
 must always be real, i.e. the listener must feel or recognise the connexion 
 of the tonal patterns with one another or with this pattern that includes 
 them all. Chap. x. 
 
 [The instructor may find it helpful at this point to offer the following 
 information : 
 
 The organ of hearing is the cochlea, in which a long thin membrane 
 occurs, called the basilar membrane. This membrane stands in connexion
 
 204 SYNOPSIS OR OUTLINES OF INSTRUCTION [CH. 
 
 with a parallel strip of sensory nerves. The cochlea is coiled up like 
 a snail's shell (without the turn at the apex of it), fig. 10. 
 
 Every note that reaches the ear comes first upon the point a of the 
 membrane, and affects a length of it that is propor- 
 tional to the length of the air- wave for that note, or 
 inversely proportional to the number of air vibrations 
 per second. Thus a note of 100 vibrations will affect 
 twice as long a strip of the membrane as one of 
 200 vibrations, but each begins to work at the 
 point a. The point of the membrane most intensely 
 affected lies at the middle of the length affected by 
 any note, and from that point to the ends of the 
 part affected the intensity probably decreases evenly. 
 
 Hence flow all the features of tones enumerated 
 in the preceding paragraphs. An attempt may Kg . 10 .^7 utline of 
 
 be made to deduce them by way of recapitula- the sensitive strip in 
 j.- a human cochlea. 
 
 The student may find such statements as thus refer to a physical 
 membrane, etc., not only acceptable and convincing, but also preferable 
 to a discussion of what we merely hear : in spite of the facts (1) that 
 (with proper guidance) tones are immediately observable by him in 
 all their aspects, (2) that he has possibly never even heard of the cochlea 
 before, and probably knows nothing of it by direct observation, and 
 (3) that, being perhaps a music student, he is eager to enjoy or even to 
 create works whose entire essence resides in sounds aswe merely hear them. 
 
 People of this age are credulous with regard to statements of a 
 scientific form regarding material things, while they will hardly even 
 believe that what they can plainly hear or see or in general have before 
 or in their minds can be properly observed or made the subject of 
 scientific study and proof. When invited to note a description of mere 
 experiences (e.g. sounds and tones) and to follow a train of reasoning 
 about them, they feel that they are being deluded or beguiled into 
 mystical fancies, if the conclusions attained embody results that surprise 
 them. But there is no reason why we should be unable to observe 
 aspects of experiences (e.g. of tones) and by reasoning to infer things 
 about them that we had not previously observed or known. 
 
 Even writers on the science of music seem not uncommonly to be 
 almost afraid to trust the verdict of the ear, as if the system of these 
 verdicts that accumulate in the course of time would likely -show itself 
 to be erratic and confused, uncontrolled by any sort of law and order.]
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 205 
 
 14. Melody is constituted not by a mere succession of tones, but 
 by a motional phenomenon or connexion between tones which supervenes 
 upon their sequence, provided their pitch difference and their distance 
 in time is not too great. This motional connexion is somewhat the 
 same as the connexion that turns the successive stationary pictures of 
 the cinema into a continuous picture in which the parts move. Many 
 useful analogies can be drawn between these melodic and cinematic 
 motions. But they must not be exaggerated. Melody is not a slurring 
 or gliding tone; it is a much finer kind of motional connexion than 
 that, and it supervenes even when the sequent tones do not glide in 
 pitch at all. 
 
 The motional connexion between tones is the more obvious the 
 nearer the tones lie to one another. The consonance of the interval 
 included in a leap probably improves the connexion. So does any other 
 influence that strongly suggests a next tone, especially a near one. 
 Hence arises, for example, the powerful tendency of the 'leading' 
 tone to the tonic. 
 
 15. As soon as music passes the homophonic stage of a single 
 melody with a rudimentary accompaniment, the chief problem is how 
 to make two or more melodies move clearly alongside one another 
 without their arresting or confusing one another's motions. This is 
 the problem of polyphony, and music in two or more voices is well 
 termed polyphonic. It might also be called polymelodic. A clearer 
 term is afforded by the word 'paraphony,' which associates the notion 
 with the familiar word 'polyphony,' but indicates specially that the 
 voices have to move alongside (para) one another, without any mutual 
 disturbance or confusion (p. 155 ff.). 
 
 Harmonic does not differ from polyphonic music, as the latter term 
 might suggest, in not satisfying the needs of paraphony and so in not 
 being a combination of voices, but in being rather a new way of putting 
 tones together. In so far as paraphony is concerned there is practically 
 no difference between the two styles. The difference lies in the aspect 
 of tone combinations that is made the ground of the artistic synthesis. 
 In polyphony the melodic outlines or figures of each voice are the 
 elements of artistic construction; each melody must be either at least 
 tense, cogent, and expressive, or at most skilfully thematised, commonly 
 in close relation to the accompanying melodies. In harmonic music 
 this interest has been largely neglected in favour of a great concern 
 for the tonal patterns created at their moments of simultaneity. These 
 patterns (or textures or colours all these terms suggest a modification
 
 206 SYNOPSIS OR OUTLINES OF INSTRUCTION [CH. 
 
 of the 'surface' of tonal masses), are now made the chief ground or 
 object of artistic workmanship. Only one voice commonly the highest 
 is thematised, the others hardly at all. At times none may be so; 
 then the music wafts along in breezes of harmonic surface or colour. 
 But the needs of paraphony must always be satisfied. And so we must 
 recognise that all music is primarily and essentially paraphonic and 
 only in the second place either polyphonic (in the historical sense) or 
 harmonic (or ultra-harmonic). Chap. xix. 
 
 16. As far as circumstances permit, simultaneous melodies form 
 themselves, or present themselves formed to our attention, just as do 
 single melodies. We do not need to attend to them and thereby to 
 combine their tones into a sequence by an inner act of mind. In fact 
 we do not know which tones belong to the one or other melody until 
 we find them incorporated in it. The melodies flow beside one another 
 like the streams of raindrops that run down a window pane in devious 
 neighbourly courses. The essential and inevitable overlapping of tonal 
 volumes does not seem to create any general disturbance of paraphony. 
 Hence we may infer that melodic motion holds rather between the 
 pitch centres of these volumes than between the volumes as a whole. 
 
 This primary freedom of paraphony is characteristic of the intervals 
 of thirds and sixths. Chap. xm. 
 
 17. The other intervals offer less freedom to paraphonic flow. In 
 relation to the series of fusions (given in paragraph 7 above) we find 
 towards the consonant end an increasing effect of (symphonic) obscurity 
 and arrest; towards the dissonant end an increase of (diaphonic) con- 
 fusion and restlessness. 'Symphony' expresses the apparent unitariness 
 of the interval's mass of sound, its approximation to the balance and 
 unity of a single tone. Compare the Greek definition of 'symphony.' 
 'Diaphony' indicates the apparent inter-penetration of the tones, by 
 which they obscure one another's pitches and outlines, and so suggest 
 a movement to other neighbouring tones that will give either greater 
 balance or easier paraphony. Symphony (and unison, i.e. a single tone 
 as the momentary identity of two voices) therefore creates a break 
 of melodic flow and a feeling of arrest, diaphony tends to obscure the 
 melodic lines and suggests transition to more peaceful terms. Chap. xvm. 
 
 18. We have already seen how the symphonies of the octave and 
 fifth are created by the peculiar way the volumes of these tones fit 
 into one another so as to appear like one balanced whole or like one 
 tone. In diaphony there is a collision or rivalry of the defining parts 
 of the volumes of tones due to their proximity to one another (thus
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 207 
 
 all the diaphonies neighbouring on the octave and on the prime or 
 unison) or to their failure by a little to yield the balance of the fifth 
 (thus the small zone of diaphony on either side of the fifth). In paraphony 
 the defining points of the two volumes stand well clear of one another 
 so that the two lines are easily apprehended as distinct even though 
 simultaneous. These relations can be traced more or less distinctly 
 throughout the greater part of the second octave, especially up to the 
 twelfth. The irregular position of the fourth in this scheme corresponds 
 with the somewhat unstable character of that interval in music. 
 Chap. xxi. 
 
 19. The effects of symphony and diaphony are, of course, worse 
 when two intervals of the same species follow one another than in a 
 single interval of that kind. Hence sequences or ' consecutives ' are 
 forbidden, the more stringently the more symphonic or diaphonic 
 the interval concerned originally is : octaves (or unisons) most, fifths 
 less, fourths only a little; minor seconds and major sevenths more 
 than major seconds and minor sevenths; or minor seventh and tritone 
 perhaps less than other dissonances. Chap. xn. 
 
 But single intervals are forbidden too, or are at least inadvisable, 
 unless they are approached in a way that counteracts or obscures their 
 own symphonic or diaphonic effect. Chap. xv. 
 
 20. Of these counteracting or obscuring methods the most important 
 is the use of contrary motion in approaching symphonies and diaphonies. 
 This has the effect of annulling the unitariness and arrest peculiar to 
 symphony and the obscurity and confusion of diaphony. The lines of 
 motional connexion, proceeding in contrary directions either inwards 
 or outwards from the pitches of one interval to those of the next, stand 
 clear of one another and are therefore paraphonic. 
 
 Oblique motion has a kindred function. The stationary voice 
 involves only a renewal of the sound already heard. The other voice 
 has then almost the same freedom of action as a single melody. There 
 is no arrest or confusion between the two voices the stationary and 
 the moving one, because the one voice moves so clearly and distinctly. 
 
 Similar motion probably leaves the intervals quite unaffected, so 
 that their own naturally inherent characteristics of unity and balance 
 (symphony) or confusion and rivalry (diaphony) emerge unmodified. 
 The reason why similar motion is so often forbidden in musical con- 
 struction is not that such motion is a bad method in itself, but because 
 it is ineffective to remove the undesirable characteristics of the intervals 
 concerned. It is the interval that is bad, not the similar motion.
 
 208 SYNOPSIS OR OUTLINES OF INSTRUCTION [CH. 
 
 Contrary motion is specially good because it annuls the confusing effects 
 of symphony and diaphony. 
 
 Melodies should not cross or overlap because the motional connexion 
 that constitutes melody tends to establish itself through the nearest or 
 easiest steps. Thus the individuality of melodies tends to be broken, 
 unless, of course, a special character is given to each by its tone blend 
 (or timbre) or by other means such as the special figures or themes 
 contained in the melodies, and so on. Chap. xx. 
 
 21. Of the tones of an interval the lower is on the whole the pre- 
 dominant one. If other tones are inserted between the first two, they 
 are less prominent than either of the others. Thus we can readily 
 establish a grading of prominence or exposure amongst the pairs of 
 voices of a polyphonic work. This grading is : B-S, B-A and B-T, S-A 
 and S-T, A-T, where the letters stand for Bass, Tenor, etc. (p. 102 f.) 
 Whatever appears in any pair of voices is the more effective in proportion 
 to the exposure given by the voices. If the effect is bad, it will be worse 
 than it otherwise would be ; if it is good, it will be better. Hence arises 
 the gradation of prohibitions according to the pair of voices to which 
 they refer. Chaps, xn, xv. 
 
 The prominence of the inner parts will naturally be less the greater 
 their total number. Hence we find a gradual relaxation of prohibitions 
 with the increase in the number of parts. Even the outer parts are 
 less prominent when there are many altogether than when -there are 
 few, so that the relaxation of prohibitions is quite general in its degree. 
 Each melodic rivulet is less clear and distinct when there are many of 
 them. The common preference for four-part music probably indicates 
 that in four parts a fair balance is attained between (a) the disadvantage 
 of the unvaried exposure of all details and the thinness of interest in 
 two or three voices, and (6) the disadvantage of the loss of distinction 
 of details and the overfeeding of the interest by more voices than four. 
 Chap. xx. 
 
 22. Contrary motion is not the only means of reducing symphony 
 and diaphony to paraphony. Any means is effective in so far as it 
 strengthens the melodic connexions leading through the unfavourable 
 interval. 
 
 An obvious device consists in making the motional tension of each 
 part specially strong. That tension, as we saw, is the greater the nearer 
 the tones lie to one another in pitch. Hence at critical moments when 
 an unparaphonic interval has to be passed without the help of contrary 
 or oblique motion, we may help it is the artist's problem whether
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 209 
 
 that help is at any time sufficient the motion by step rather than by 
 leap in one voice or in both. As the highest voice is in harmonic music 
 usually the most distinctively melodic (or thematic), the step will be 
 more effective in it. Here again we can easily set up a gradation of cases : 
 worst leaps in both voices, step in lowest and leap in highest, step 
 in highest and leap in lowest, steps in both voices best. If one of the 
 voices be an inner voice, the last degree of this series would probably 
 be an unnecessary embellishment (p. 128 f .). 
 
 23. The two preceding paragraphs (21, 22) indicate that neither 
 paraphony nor polyphony is to be supposed to be limited to two voices 
 or to any definite number of them. The problems of paraphony run 
 continuously into any number of parts. The effects are complicated 
 only by the greater number of intervals that then appear and the 
 difficulty of regulating them all together to the desired effect, either 
 melodic motion for a specified length of time or arrest of it at a certain 
 time or a variable degree of this arrestive effect. The difficulty of 
 regulating many parts at once is due to the different symphonies and 
 diaphonies that inevitably appear when a column of paraphonic intervals 
 is formed, e.g. ceg or ce^g, etc. (We suppose here logically that polyphony 
 sets out from an attempt to combine several paraphonic intervals 
 together. Really the general course of progress went probably rather 
 from the greater consonances, which first drew the greater attention 
 to themselves and were used to embellish a single melody during its 
 course and without themselves forming another melody. The slowness 
 of early progress was probably largely due to the obstruction natural 
 to this starting point. The consonances are the easiest intervals to find 
 but the more difficult to use artistically.) 
 
 The resultant effect of such combination depends upon factors 
 already enumerated, especially the nature of the interval formed and 
 the exposure given to it by the voices it stands in. Thus we can arrange 
 the various chords that involve the same set of pitch-names according 
 to the greater stability of effect. For example, of chords involving 
 the tones c, e, and g, ceg is the stablest and most arrestive, because 
 it contains a fifth (consonance of second grade) in the outer voices 
 and the most fused paraphony (major third) in the outer voices. Of 
 the two inversions the second gc^e 1 is better than the first egc 1 , because 
 the former has major paraphonies for the latter's minor ones and the 
 fourth is less noticeable in the upper voices. Similarly in chords con- 
 taining the tones 6, d and / the worst or most confusing arrangement 
 is bd l f l , thenfbd 1 , while dfb is the best. 
 
 W. F. M. 1*
 
 210 SYNOPSIS OR OUTLINES OF INSTRUCTION [OH. 
 
 24. Because of the general trend of these differences and also because 
 of the theoretical interest of columns of thirds (major or minor), the 
 arrangement of the tones of a chord that gives a column of thirds 
 has been called its root-position. We can readily think into that term 
 the notion of greater stability, arrestiveness, or paraph ony. But it 
 is wrong to suppose that any chord has been in any sense generated 
 from any one tone, as the older theories that operated exclusively with 
 harmonics supposed. But a small chord, e.g. bd 1 / 1 may well come to 
 be felt to be a part of another larger chord (gbd 1 / 1 ) in so far as the latter 
 (or its 'pattern') includes the former and comes to be so important 
 for music generally as to attach the former to itself as a mere part 
 or reminiscence. But this connexion, if it is to be valid, must be really 
 felt. 
 
 When symphonic intervals are combined with diaphonic, we find 
 that the loss of distinction due to the great unitariness of the former 
 is largely annulled by the loss of distinction due to the confusion intro- 
 duced by the latter. Consequently single and second symphonies (octaves 
 or fifths) may pass unguarded (without the help of contrary motion, etc.) 
 in discords. 
 
 25. A special paraphonic difficulty is created by the interval of 
 the fourth, especially when it is exposed by resting upon the centre 
 of a sound mass the bass. The fourth is not a dissonance, as has 
 often been asserted. It is what we already know it to be the lowest 
 grade of symphony. But especially when it is exposed rhythmically as 
 well as phonically, it seems to call for or to suggest the neighbouring 
 major third, and so to claim a resolution, as do the dissonances. If this 
 suggestion is to be avoided, exposure of the interval must be reduced 
 as far as possible, and the melodic line leading through the chord in 
 which the fourth appears must be made as cogent as possible, especially 
 the line through the bass. If the suggestion towards the third is welcome, 
 it is increased by rhythmical exposure and so a special progression a 
 (symphonic) cadence (or general arrest of melodic flow) is attained. 
 Chaps, xvi, xx. 
 
 The case of the bass-fourth is probably only one of a class of similar 
 cases. Another example seems to be given by the minor sixth, which 
 has been held to be dissonance, although it is now generally treated 
 as a consonance. But the interval identical with it in equal temperament 
 the augmented fifth is always a dissonance, probably because the 
 augmented triad so strongly suggests the major triad that the former 
 appears as a distortion of the latter.
 
 211 
 
 This reason for the difficulty of the fourth from the bass explains 
 why the effect of such a fourth is not obliterated in a discord, while a 
 second fourth in the same voices (consecutive fourths) is relieved, if 
 either the second (or even the first) stands in a discord. In the latter 
 case the fourth acts merely as the symphony it is ; in the former its effect 
 is due to the call for the major third excited by its exposure on the bass. 
 
 26. The tendencies and possibilities of resolution of dissonances are 
 not so much the result of any sort of connexion between different 
 intervals, as far more probably a general expression of the most cogent 
 melodic movements that will satisfy the propulsion of melodic move- 
 ment that is characteristic of diaphony. The most cogent movement is 
 in the first place the shortest (in difference of pitch), but due regard 
 must be paid to the limitations imposed by key structure, the paraphonic 
 nature of the resulting (following) interval or of the whole chord in 
 which it stands, thematic form, change of key, etc. Apart from these 
 determining conditions there are probably no limits to the possibilities 
 of resolution. 
 
 27. It is not to be supposed that chords are essentially different 
 from intervals. Both of them are in music primarily paraphonic 
 structures, which differ only in the number of voices involved. The 
 putting together of three voices is only an extension of the task of 
 combining two melodies, not a new kind of task. But we learn to look 
 upon the structures formed by three or more voices as typical volumic 
 patterns, which we reduce to a few comprehensive types. The relation 
 of inversion, for example, forms a very important step in this reduction. 
 So accustomed do we become to this procedure that we largely cease 
 to be able to look upon intervals (of two tones) except as fragments 
 of chords. Nevertheless there was a time (for music generally and for 
 each musical person) when intervals had not yet come into these 
 relations. The final result of long experience with the most frequent 
 types of chords is the formation of a system of chords and their relations 
 and progressions, commonly known as Harmony. 
 
 The artistic concentration of attention upon harmonic rather than 
 upon melodic form and relations leads to a special systematisation of 
 chords by means of which they are grouped round a centre (tonality) 
 with parallel sub-centres (keys). Tonality reacts upon the scale, moulding 
 it so as to make it yield the biggest and most coherent system of chords 
 possible. From this diatonic harmony the system is extended gradually 
 to a system of chromatic harmony and to other types that are at present 
 insufficiently defined. 
 
 142
 
 212 SYNOPSIS OK OUTLINES OF INSTRUCTION [CH. 
 
 28. The development of tonality raises into importance the chords 
 that are related to one another in certain special ways, especially 
 those that stand at the interval of a fifth or fourth from one another. 
 Here we have in a new form the relation of balance characteristic of 
 the symphonies; the octave virtually drops out because the repetition 
 of a chord an octave away is in all but its 'brightness' or volumic 
 size the same chord: or the octave is just the greatest balance, if you 
 like. The fifth (or fourth) is the next, and therefore the main, pivot 
 of all tonality. These points of tonality are also the only ones that 
 repeat a chord exactly without any distortion. (Possibly this balance 
 of chordal relations at the fifths is responsible for the final form of the 
 scale.) So we come to think these points easily together. And their 
 connexions can thus combine with the circumstances already indicated 
 to counteract the effects of symphonic and diaphonic intervals towards 
 any special end. 
 
 There is some evidence that this power of harmonic connexion 
 varies with its closeness, being greatest for the fifth-fourth, next for 
 the third-sixth, and least for the second-seventh (or no) relation. But 
 the matter probably requires further justification before it can be taken 
 in this general form. For the first grade of harmonic connexion there 
 is no doubt of the existence of a modifying power. 
 
 29. It must not be thought that the laws of primitive music are 
 annulled by its development. They are rather merely complicated and 
 fulfilled. Rules of harmony that bind one age are not disregarded in 
 the next. Nor do we merely learn to tolerate what a previous age 
 forbade. We rather learn to annul or to transmute the effects that 
 once created strain, by bringing to bear upon them influences that the 
 previous age had not discovered, but that are essentially the natural 
 products of musical development, or, better, of tonal artistry. We 
 bring the once jarring phase into a setting in which it forms a proper 
 extension of those very laws that were previously the just ground and 
 reason of its prohibition. Music is an orderly realm of beauty, intelligible 
 in every aspect; it is not a temporary code of arbitrary preferences 
 that extol themselves as art. 
 
 The main thesis propounded in this book, so far as the actual 
 structure of music itself is concerned, is embodied in the paragraphs 
 of the preceding summary from no. 14 onwards. It may be briefly 
 stated here : 
 
 The fount and origin of probably all music whatever is melodic
 
 xxn] SYNOPSIS OR OUTLINES OF INSTRUCTION 213 
 
 movement, or simply, melody (not limited, however, to thematised 
 melody). Primitive music is monomelodic, embellished by reduplication 
 at some interval or with irregular heterophonic accompaniment. Modern 
 music, whether it be classed as polyphonic or harmonic, belongs to a 
 great era of polyphony, of which the essential problem is the construction 
 of concurrent melodic streams that will leave each other's motions 
 unimpaired and produce effects of arrest as they may be desired. The 
 solution of this problem embraces the solution of the nature of similar, 
 oblique, and contrary motion, of the prohibition of consecutive intervals 
 of the same kind as well as of 'hidden' consecutives, of the puzzle of 
 the fourth from the bass, of chordal structure in general and of the 
 differences from one another of a chord and its inversions, of the need 
 for resolution of discords and its tendencies and possibilities, etc. 
 
 Or : it is commonly held that our music is "the concord of sweet 
 sounds," a structure of which the fundamental material and ornament 
 is consonance in its special or more perfect grades. We might rather 
 say that music is a 'concourse' of sweet sounds, in the literal sense of 
 the word, a structure in which movement or melody is the fundamental 
 material. There is rather an opposition than a kinship between the 
 functions of melody and of consonance. The latter is the principle 
 of arrest, inimical to the free course of melody. Music has been created 
 rather in spite of consonance than by its help. 
 
 Thus the Greek definition of consonance is upheld both in its connota- 
 tion and denotation. Symphony reveals a loss of distinction of tones 
 in apparent unity. The neutral intervals of thirds and sixths must be 
 recognised as paraphonic; their tones are freely distinguishable. In 
 dissonance or diaphony we find a loss of distinction in confusion or 
 harsh collision. Thus we give new significance and confirmation to 
 the classification of intervals proposed by Gaudentius. Music progresses 
 by greater differentiation between intervals and by the discovery of 
 further factors that modify their paraphonic properties, not by a 
 gradual lowering of the border line of consonance.
 
 CHAPTER XXIII 
 
 THE OBJECTIVITY OF BEAUTY 
 
 REMARKS have been made at various points throughout the preceding 
 chapters in deprecation of the common tendency to look upon beauty 
 as a matter of convention, or of merely subjective personal fancy, 
 such as prevails in one person or age and may be rejected in another. 
 One feels a certain reluctance towards passing from the study of the 
 foundations of musical beauty in detail to the exposition or defence 
 of a general thesis such as that of the objectivity of beauty. Those who 
 study the detail closely will surely come of themselves upon the general 
 point of view, and will feel its cogency. And there is a great pleasure 
 in thinking out such conclusions and in seeing how they extend their 
 satisfying graces into the whole of one's world. On the other hand 
 there is a certain amount of laborious thought involved in these visions, 
 the communication of which may help to lighten the work of others. 
 In any case, every one of us has his own particular world to fill with 
 strength and joy, so that there is no real risk of robbing any man of 
 the pleasant excitement of that task. 
 
 The age we live in is in many ways strongly coloured by the outlook 
 of subjectivism. We have passed through the time in which the first 
 great generalisations of biology were made. Each type of creature has 
 been shown to be the product of innumerable influences that not only 
 work upon it now, encouraging one individual and eliminating the 
 other, but that played in the same way upon the infinite series of its 
 ancestors. The choosings of chance have made us as similar or as 
 different as we actually are. Our members, their forms and functions, 
 our innermost structure and processes, in short every infinitesimal part 
 of us is to a greater or less degree peculiar to us, unique, sifted off from 
 the great drift of life's variations, and stamped 'individual.' How, 
 then, can we expect to be so fundamentally akin that the world should 
 be the same even for two of us? Why, our very eyes, our brain, and 
 the minute chemistry that is its change, have been selected from myriads 
 of possibilities! Even beauty has been held to be merely contributory 
 to life's functions, a subjective accident enlisted in the work of
 
 CH. xxm] THE OBJECTIVITY OF BEAUTY 215 
 
 reproduction to be a symbol of the fitness of male and of female to one 
 another. And we make our arts out of these symbols by wandering 
 from the straight path of life's purposes and by perverting a good 
 function to a useless play. So we come to look upon art as perversion 
 and upon the artist as an unpractical trifler, stifling the true functions 
 of his mind which should serve the great ends of life by combining 
 them into vain display. 
 
 But the artist reaps the inevitable reward of his perversion. He 
 flatters himself with the thought of progress : in the end so many 
 revolutions are wrought in his art that the nothingness and unreality 
 of all of them become too obvious to ignore. Art then comes to be 
 the whim of an age, the latest device to thrill and to dazzle a mind 
 that is 'real' only when it passively acts as a drop in the incoherent 
 drift of life, but that, blowing itself out to some semblance of intrinsic 
 worth and orderliness, shows a brief iridescence of meaningless changes 
 and bursts into dust. 
 
 And then the ruin of art begins to infect the other parts of mind. 
 Thought is even degraded to be a mere scheme of symbols, useful for 
 our practical life, but worthless in themselves. We are shut up within 
 the impenetrable wall of a senseless 'mind' without any means of testing 
 the true worth of the bonds that seem to link us to a world beyond, 
 and with every 'reason' to suppose them illusory. Morality becomes 
 the preferences of a social group, and the world-race of nations is then 
 to the strongest, who grants himself all the sanctions of subjectivism. 
 In the vast struggle of these years we are not striving merely to make 
 room for every man's illusions against one tyrant's ambitions. That 
 would really yield the senseless chaos that freedom seems to the enemy 
 to be. We are rather labouring to fulfil the demands for obedience 
 of laws that claim objective worth and that promise to give our souls 
 the dignity of reality, of being members in a coherent intelligible world. 
 
 Another source of subjectivism was created by the critical philosophy 
 that has been lauded for a century and a half as the greatest revelation 
 achieved by thought. It is true we must examine the work of knowledge 
 with all care to see what it can validly establish. But if our conclusion 
 be that true knowledge is so overlaid with the devices of the mind itself 
 that it is impossible to say anything about the material upon which it 
 started to build, we are driven therewith upon the rocks of subjectivism, 
 no matter how zealously we repeat to ourselves : "what the mind once 
 claimed to know (really), it may still claim to know (virtually)." 
 
 Of course we are glad to have cast away the crude illusions of know-
 
 216 THE OBJECTIVITY OF BEAUTY [CH. 
 
 ledge that always bind our thoughts so long as we do not examine 
 evidences carefully and reason with caution. In a sense, however, 
 these illusions are only the natural result of a force of knowledge that 
 is eager to work and has not yet learnt to harness itself to the ideals 
 of completeness and security. But if our critical study of knowledge 
 is to lead us to suppose that thought itself is a deceit like a kaleidoscope, 
 revealing nothing beyond its own gyrations but the existence of light 
 itself, then have we cause for misery. We shall be prone to think that 
 the older 'dogmatisms' that knew nothing of critical scepticism, were 
 really sounder products of the spirit than the critical philosophy itself. 
 They may have known all that we know of evidence, consistency, 
 and system, although they were greatly deficient in facts and left in 
 their expositions too much to the imagination and understanding of 
 their readers. 
 
 At any rate a sounder vigour is beginning to show itself openly in 
 the sciences of biology and in philosophy. We now see that chance 
 of itself does nothing, and that selection merely selects or favours one 
 of many possibilities, each of which is founded upon the coherence of 
 objective forces that we call laws of nature. All the possibilities of 
 inanimate or animate combination are realised, if only the forces by 
 which they are surrounded tolerate them ; and they endure in proportion 
 to their stability and coherence. Every individual, no matter how 
 special or 'rare' he be, is founded upon the not innumerable forces 
 that make an infinite variety of combinations possible. His unique 
 composition involves only the limited gamut of the world's scale. In 
 respect of generality and immutability the underlying laws of life take 
 rank beside even the laws that rule the great bodies of the universe. 
 These are the great chimes that tell the years, those are the fleeting 
 colours of an orchestral hour. But they both go back to the primary 
 balance and beauty of pure tone. 
 
 Our study of music has shown us that order and systematic relation- 
 ship are the ground and basis of the art, not capricious selection and 
 convention. We have indeed to learn to combine the fundamental 
 forces of tone; but there is no caprice or chance in the forms that 
 please. Chance pertains only to the experiments that we shall at any 
 time make, within a certain scope allowed by the age in which we live. 
 We cannot yet adopt what may seem beautiful to a later age. Not that 
 we may not already see some beauty in it; but the orderly scheme of 
 beauty we have thus far gathered does not yet offer it any welcome.
 
 xxm] THE OBJECTIVITY OF BEAUTY 217 
 
 No place has been prepared for it within the art. Music has to grow 
 till it is able to comprehend each special beauty as its own, just as the 
 mind grows gradually towards the wisdom of its years. But both the 
 beauty and the wisdom of a later day are the true fulfilment of their 
 earlier forms, children of the same elemental order and purpose. 
 
 It is evident enough that there is a structure and necessity in what 
 we find to be beautiful. That is true in broad outline at least, however 
 difficult it still may be to carry the demonstration down to the finest 
 details. Beauty is of our creating only in the same sense that the forms 
 of life are the result of selection. It comes before us and we find it good. 
 We combine its simpler parts as we can, and we then find laws in the 
 conjunctions that please us. We use for art a material that has already 
 been wrought from the ore, as it were. In short we discover beauty 
 just as we discover truth or even goodness. 
 
 Beauty is not beauty merely because we feel pleasure with it. It 
 is true we should not respond to it or feel it but for pleasure. But that 
 is really a tautology. It means only that if we did not respond to it, 
 we should indeed not do so. On the other hand, were beauty not what 
 it is, it would excite no feeling in us. And mere feeling of a false kind 
 never makes beauty. We have to adjust our souls so as to feel aright, 
 just as we have to be careful to think truly. Feeling is merely the 
 thrill that beauty strikes through our being, a symbol of our fitness 
 to share its high functions. We are aware at such times that our 
 experience is most cogent and coherent, as it is likewise in true thought 
 and deed. 
 
 Moreover, beauty reaches back continuously into the physical world. 
 The tonal volumes and pitches that are our own experiences, find an 
 objective counterpart or parallel in the excitations spread over the 
 sensitive surface of the ear. And the simplest parts of these at least 
 depend upon equally regular processes in the aerial medium. Although 
 the three stages of the physical, physiological, and psychical are not 
 precisely alike in all respects, yet their differences are completely 
 comprehensible and regular. Except in so far as we cannot yet discern 
 the inherent continuity of mind and matter, there is no mystery in their 
 general continuity of action, as far as hearing at least is concerned. 
 What comes to us as beauty is, therefore, founded upon the order and 
 regularity of the natural world. In fact it is itself a part of the natural 
 world. 
 
 If anyone cares to read this conclusion as "beauty is physical," 
 no great objection could be urged, so long as the meaning implied is
 
 218 THE OBJECTIVITY OF BEAUTY [CH. 
 
 not altered to suggest that we understand the continuity running 
 through matter, body, and this 'physical' mind. That we manifestly 
 do not yet understand. If our comprehension is to be secured by setting 
 down the realistic constructions of ordinary knowledge and of science 
 as mere classifications of phenomena or as mere symbols or counters 
 for thought, we are surely on a false track. No one is really inclined 
 thus to sacrifice the results of science as illusory. Nor has anyone 
 succeeded in basing what we know in science in the broadest sense upon 
 what is exclusively phenomenal to us, upon sensations, for example, 
 and upon nothing else. Nor has anyone yet shown that the entities 
 posited by science are essentially of the nature of the phenomenal 
 units of our sensory experience or the like. What the elements and 
 complex units of the physical world really are in themselves apart 
 from their order and regular changes, we do not know. If we pretend 
 to know by trying to believe they are all illusory or fictitious ('reasonably 
 fictitious,' of course), we only deceive ourselves. For we can never 
 really believe that. The well ordered panorama, all of which to the 
 last particle we can see with direct gaze, that some would have us 
 believe in, is a myth. We cannot in ourselves be all-seeing and all- wise. 
 And if we see only in part, and that with great labour of concentration, 
 then there is much of the world that reveals its movements and forms 
 to us only through our own very limited panorama of phenomena. 
 Thus there must be many things in the world whose inner being remains 
 indefinitely hidden from us. Beauty may be a part of the physical 
 world, if you like, continuous in action with it; but it is not a part of 
 that real world which we call physical and whose inner essences are 
 hidden from us. For we have the full being of beauty before us, however 
 difficult it may be for us to understand it completely. We feel it entirely ; 
 it is there before us in all its native essence. 
 
 If beauty is thus objective, a part of the natural world, we have a 
 greater satisfaction. What we have before us is the promise of all. 
 Beauty then belongs essentially to the world. And the world we may 
 look upon as really so beautiful as we naively observe it to be. Its 
 charms are not the mere dreams of a creature of chance, dreams that 
 will vanish with its death. We may be sure that, wherever there is a 
 mind to view the sunset or to hear a song, provided that mind has 
 grown to the stature of the task, there beauty will be evident. There 
 is not one world for you and another for me, but one according to 
 our varying powers of apprehension for all.
 
 xxm] THE OBJECTIVITY OF BEAUTY 219 
 
 In pre-critical days every discovery of awakening science was apt 
 to be hailed as new evidence of design in nature and of the presence 
 of God's hand in the world. We should not generally now be so simple 
 as to imagine that law and order form a sufficient proof of dependence 
 upon an omnipotent person. But for all that we cannot forgo our 
 wonder and admiration at the awful complexity and the marvellous 
 stability of things. No amount of belief in mechanism will render us 
 quite cold to these effects. 
 
 We may indeed tell ourselves that our minds, having been evolved 
 in dread of nature's rough forces and of the savagery of beast and man, 
 are only too prone to be moved by what is vast and great and complex. 
 And we may dispassionately infer that our notion of a natural body 
 is as of a thing we might ourselves have tried to make a machine 
 and its works; and that we pass therefrom merely by association of 
 ideas to a notion of the effort and ingenuity that the works of nature 
 would have involved, had a mind like ours devised them. From this 
 point of view we read the effusions of the earlier philosophers of nature 
 with a certain amount of condescension. 
 
 We are supported in this attitude by the reflexion that though 
 mind itself may feel orderly and lawful, such a feeling is but illusion. 
 Really it is merely subjective and accidental, a product of the subtlest 
 cerebral chemistry or physics, not only worthless as a sign of manifest 
 order, but obviously quite beyond any hope or possibility of analysis. 
 We can take as our test of mind's action no formation inherent in itself, 
 no continuity of process that binds it to an objective world, but only 
 the success of its functions, which in its turn is evidenced only by 
 the mind's tolerance of them. Not only can the existence of a universal 
 world not be proved by mind, but we must finally admit that whatever 
 the mind is content to accept is acceptable (or 'true'). There is not 
 only no permanent world from which the world might seem to acquire 
 order by merely reflecting it, but the mind itself cannot be expected 
 to generate order. Its autonomy is sheer heteronomy, its sanctions 
 are its own acts, its only law is mere function. 
 
 But if law and order are discovered and proved for mind itself, what 
 then? We may expect to hear the reply : what you have shown to be 
 so orderly is not mind but nature ; or, you have merely carried through 
 a useful, helpful scheme, which has no further validity than the brevity 
 and expository value you have shown it to possess. Both of these 
 interpretations will surely in the end prove tiresome and nauseous. 
 For as we all suppose ourselves to have some sort of mind of our own,
 
 220 THE OBJECTIVITY OF BEAUTY [CH. xxm 
 
 we shall hardly consent to abandon it all to nature; nor can we enjoy 
 the jest of a pragmatic world for ever. We shall gradually recognise 
 that the order of as much ' mind ' as we at any time understand is akin 
 to the great * design ' we now see in nature. We shall become convinced 
 that 'mind' is of a piece with nature. A great and universal objectivity 
 runs through both. 
 
 Shall we then be more inclined to see mind in nature, a vast 
 thought, actually delineated by our sciences, 'thinking' itself (or 
 whatever other faculty of mind we may conceive it to be), forwards 
 by its rotations and syntheses in its own vast sensorium (or conceptorium 
 or effectorium), although we can in no way discern the quality of 
 consciousness that pervades it? Or shall we humbly infer that our own 
 minds are a mere mechanism too, rolling on because it once got started 
 on its futile gyrations, endowed with a stability that is a synonym 
 for 'it is what it is'? Shall we not rather learn to see that, in mind, 
 not only is the essence of order and of law actually made manifest, but 
 inner coherence and stability has even become the ideal of all its effort 
 (pervading the great faculties of sense-beauty, concept-truth, and 
 emotion- joy, love, goodness)? When these ideals seem attainable, its 
 very being is rapture; it is misery so long as they seem indefinitely 
 elusive. 
 
 Whatever our general argument or our detail of knowledge may be, 
 we shall probably all incline always to some sort of an idealism, however 
 much of reality we may at the same time find around us whose kinship 
 of being with us we cannot fully prove, at least as regards the last 
 essential phase of that proof. And we may do well to be so inclined. 
 It is not to be expected that a creature of this world should be willing 
 to sacrifice the only direct vision of being it possesses for any of the 
 forms of other beings that its knowledge may describe. Until we learn 
 that mere form can constitute a being, we shall hope in our modesty 
 that the things of this world are essentially (or in their inherent being) 
 greater (spirits) than we are, and in our pride that they have in themselves 
 some portion of the insight of mind and some glow of the emotion we 
 find in ourselves. 
 
 The knowledge we have won may at least make us hope and believe 
 that in beauty we are in direct touch with a real aspect of the world. 
 And if the world appears to us in the order of law and in the harmony 
 of beauty, why not also in the kingdom of goodness and love?
 
 CHAPTER XXIV 
 
 AESTHETICS AS A PURE SCIENCE 
 
 AESTHETICS as a pure science is simply a part of the pure science of 
 psychology. So all the characteristic features of the latter notion 
 appertain to the former. 
 
 Psychology is concerned with the parts and the structure of what 
 is commonly called 'mind.' That includes, according to the doctrines 
 of modern science, not only such things as memory, thought, and 
 emotion, not only the bodily feelings of hunger, pain, warmth, posture, 
 and movement, but also all sights and sounds, distances, forms and 
 spaces. Of course some folks object to having these last things called 
 sensations and experiences. But that does not matter at all for the 
 present, so long as we have good reason to group them together and with 
 the bodily feelings just mentioned. All these things call for systematic 
 study together. 
 
 In fact, they seem to form a distinct world by themselves and apart 
 from all that science can tell us of the physical forces that arouse pain, 
 warmth, sound, sight, or of physical form and space. The ordinary 
 man lives in his world of sensory feeling as he finds it; he does not 
 concern himself with the physics of sound, light, or warmth, or even 
 of space. Psychology is the name of the science that has to arrange 
 and to explain the parts of this world in relation to one another, however 
 it may afterwards (or otherwise) join hands with any other science 
 (say physiology or physics) to show how feelings depend upon the body 
 or upon matter. This systematic work of psychology has to be done 
 without filling up gaps of ignorance by disquisitions on the relation of 
 the feelings to the body or to matter, and without giving theories about 
 feelings that are merely inferences from what we know of matter or of 
 body (unless the temporary deficiency of our knowledge makes such 
 an effort inevitable). That is what is meant by speaking of pure 
 psychology. 
 
 The work of pure aesthetics is similar to give a full description, 
 systematic arrangement, and so an explanation of every work of art 
 in so far as it is directly before the spectator's (or artist's) mind without 
 any regard for all the (otherwise so important) facts and questions
 
 222 AESTHETICS AS A PURE SCIENCE [OH. 
 
 relating to the work of art as a physical thing (marble, canvas, paint, etc.) 
 or to its effect upon the eye, the ear, the nerves, the brain. 
 
 It. has taken science very long to find out that the brain is the organ 
 of mind, so to speak. Most people have still only the vaguest idea of 
 the connexion between the two. But this ignorance in no way alters 
 the efficiency of their minds. They can observe their impressions and 
 the workings of their minds quite clearly; they know whether they 
 feel pleased or displeased, with reason or for some motive, whether 
 they are sure of their conclusions or uncertain, whether their conduct 
 seems to them good, indifferent, or disgraceful enough to need conceal- 
 ment. Of course, they do not therefore know everything about their 
 minds; they may deceive themselves in feeling, knowledge, and 
 conscience ; but yet we all know and believe that if they set their minds 
 more vigorously and honestly to work, they might improve its action 
 indefinitely. In the same way the finest art was in existence long before 
 men knew precisely of what marble and paint consist or how musical 
 instruments produce their tones. A composer accepts or rejects a tone 
 for a melody or a chord for a progression, not because he knows some- 
 thing of its physical nature, not even because he knows what partial 
 tones it contains or implies, but merely because it is beautiful and fits 
 beautifully into the work at the point in question. He has all the 
 material directly before his mind by which to judge of this without 
 needing any scientific knowledge at all. 
 
 It is the task of a pure psychology in general and of aesthetics in 
 particular to give as connected an account as possible of all that thus 
 goes on before and in the mind. 
 
 There are many, however, who say that though much can be found 
 out about the things of the mind, the account can never be made 
 complete or closed. That we shall see in the course of time, if we try. 
 We have at least a great promise of approximate completeness; for 
 do we not expect the work of the mind and of art to grow more and 
 more consistent and perfect? So, to begin with, we have perhaps even 
 more cause to believe in the uniformity and inherent consistency of 
 mind than of nature. Whether mind is in some region or process chaotic, 
 we shall have cause to believe if we fail utterly to comprehend some 
 portion of its ways. 
 
 But there are mysteries both in matter and in mind. The darkest 
 mystery of the physical world is why or how it ever was created and 
 set agoing. The very question is a sort of nightmare, through which
 
 xxiv] AESTHETICS AS A PURE SCIENCE 223 
 
 we help ourselves by the recital of beautiful myths. The problem of 
 the origin of mind is just as baffling. Science evades these ultimate 
 issues. Its only concern is to reduce the world or mind to its simplest 
 foundations. But, given these and what it learns of their nature in the 
 course of analysis, it hopes to show how the complex world and the 
 mind we know result therefrom, without the assumption of any inter- 
 ference 'from without.' We do not now suppose that God steps in, as 
 occasion demands, to wind up the clock of the physical universe or to 
 mend its works. Nor need we suppose, once the elements of mind 
 have been given by the brain it is dependent upon, and the process of 
 complication and interaction of these elements has begun, that the 
 brain here and there exerts its influence anew to keep the process going. 
 
 This last idea may be illustrated in a crude way by reference to the 
 problem of consonance. Instead of supposing that there is something 
 about the mass of sound constituted by two tones an octave apart 
 or by the common major chord that we can discover, describe, and set 
 up as the object or cause of the pleasant feeling we have in connexion 
 with it, some folks speak as if we heard the two or three tones precisely 
 as we hear each of them separately (only, of course, now together), 
 as if there were equally little or nothing in either case to account for 
 any unpleasantness, and as if the brain, so to speak, wired a separate 
 message to the mind in the code- word 'pleasant feeling,' which being 
 somehow decoded by the mind would mean : "the tone-group dispatched 
 by separate route is to be 'pleasant,' i.e. runs nicely through my nerve 
 centres." 
 
 If this sort of thing were true, the only problem of psychology 
 would be to make a rapid catalogue of the brain's code-words, and to 
 try by all sorts of experiments to catch the brain in the very act of 
 sending off a code- word and to see what was going on in it at the moment 
 (that was nice or nasty). And as a matter of fact that is exactly what 
 the program of psychology involves according to the present notions 
 of (probably) the majority of its professed exponents. 
 
 Whatever limits lurk hidden in the notion of pure psychology or 
 aesthetics, that kind of idea must surely be a travesty of mind and 
 art as we know, or, at least, 'feel,' them. It is absurd to suppose that 
 our minds are the mere puppets of a brain-and-matter show. If that is 
 the sort of world we live in, the sooner the silly play is over the better. 
 The opposite extreme is far more worthy of the vast and orderly cosmos 
 we know the world to be, at least the world of nature. Let us suppose 
 that in the mind we shall find the greatest inner order, coherence,
 
 224 AESTHETICS AS A PURE SCIENCE [CH. 
 
 self-sufficiency, completeness that is anywhere detectable at all. Let 
 us never rest till we have gone far towards realising this ideal by the 
 most stringent methods of science. And let us then learn to see the 
 great world in the light of the little mind we know, permeated and 
 bound fast by inner forces like those that call us forwards, truth, 
 beauty, goodness, although we know the stability of these cosmic 
 aspirations only in their outer forms, not at all in their inherent essence 
 or 'spirit.' 
 
 The pure psychology that has been expounded in the preceding 
 chapters is at the same time, in so far as we have confined our attention 
 to the foundations of music, a pure aesthetics. We have found a basis 
 for the art that at no point transgresses the range of what is before 
 the mind of the creative or appreciative musician. No gap in our 
 knowledge of the foundations of music, as we hear and enjoy it, has 
 been filled by our knowledge of the physical processes of sound or of the 
 bodily functions of hearing. 
 
 Of course the results attained are in considerable part new; they 
 do not embody at all what the musician may himself have thought of 
 music. But that is no real difficulty. Many an appreciative listener 
 enjoys a new work of music without being able to analyse it or to say 
 of what parts it consists; and yet no one would suggest that the work 
 was not completely before, or accessible to, his mind. In the knowledge 
 of a thing there are two important parts the knowledge and the thing. 
 And two relations of time are possible between them : they may come 
 into existence practically at the same time, or the thing may precede 
 the knowledge by an indefinite period. This latter alternative holds 
 not only for the objects of nature, but in some cases for the things 
 of the mind as well. It is a familiar fact that various difference-tones 
 have been discovered only comparatively recently; and, before the 
 time of Tartini or thereby, these were quite unknown. Similarly partial 
 tones were not heard and known, but only heard, to be in timbre before 
 Helmholtz's time 1 . And if my exposition is on the right lines, I may 
 claim to the best of my knowledge and belief that no one knew 
 
 1 In this sentence special emphasis should be laid upon the word timbre. Gevaert 
 (16, us f .. 153 1 .) is convinced that the existence of the first harmonic was known to Aristotle. 
 Cf. Gevaert's translation (16, is): "Pourquoi dans la consonance d' octave le grave repro- 
 duit-il 1'intonation de 1'aigu, tandis que 1'aigu ne reproduit pas 1'intonation du grave?.... 
 Et (en effet) la corde hypate, au moyen de sa division (en deux parties egales), produit 
 deux netes distinctes." "Does not Aristotle in his treatise on the Soul say: rj tfxavr) 
 (tone is a sort of chord)," (16, 119)?
 
 xxiv] AESTHETICS AS A PURE SCIENCE 225 
 
 before what tone and interval in their various aspects really are, although 
 every musical mind must have felt what they are a countless number 
 of times previously. 
 
 Our study of the foundations of music has now brought us clearly 
 within the range of the musician's present interests. It is unnecessary 
 for him to know precisely the nature of tone or of interval in their 
 simpler aspects, or to understand the nature of the blend of tone (timbre). 
 But no musician can produce good work without knowing, as well as 
 feeling, the effects of the various intervals upon the flow of his melodies 
 or harmonies. The analysis of these things is the lore of the music 
 student that we have freely used as a basis of induction. By connecting 
 the results of induction with the psychology of tone and interval in 
 their simplest aspects we have made it possible for the musician to 
 follow the growth of music from its foundations in mere tone up to the 
 strands and links that bind tones into coherent music. The study can 
 now be taken over by the musician himself and pursued without the 
 further help of psychology, in so far as this is the general and systematic 
 study of all kinds of sensations and experiences. 
 
 No doubt there is much still to be done, the results of which may 
 reflect greatly upon the disposition of the ground thus brought under 
 control. But the musician should no longer feel that he has no firm 
 place upon which to build the systematic structures that seem to be 
 needed for the housing of his musical analyses. We have now a clearer 
 view of what music really is; and we can venture to say that we now 
 understand much of what heretofore seemed to be so mysterious in 
 the art. In the following we shall review a few of these old problems 
 of music that are at the same time special forms of the general problems 
 of aesthetics. 
 
 There is a special ease and suggestiveness in claiming for music 
 the need for a pure aesthetics. It has been asserted in a much quoted 
 sentence that "all art tends to the condition of music" (quoted from 
 33, 227). By this is meant that music is an art that is obviously pure 
 and disinterested, and that the other arts can only with difficulty 
 reach the level of purity that is natural and almost inevitable in music. 
 But all arts are inherently pure; they all are only 'for art's sake.' 
 
 Probably the chief element in this distinction is the fact that, while 
 it is very difficult to exclude from the arts of sight the representative 
 functions of vision, there are so few representative processes in tonal 
 sound that the arts of sound remain wholly unrepresentative. How few 
 
 W. F. M. 16
 
 226 AESTHETICS AS A PURE SCIENCE [CH. 
 
 of the things around us make a tuneful noise! Writers on the theory 
 of program music (e.g. 39; cf. 42) have been able to compile only a 
 brief list : the cuckoo and the cock are alone unmistakeable ; the 
 nightingale, the thrush, and some others sing in a way that can be 
 imitated and recognised with fair ease; but the sounds of most things 
 are really noises that are so hard to imitate on musical instruments that 
 their recognition cannot be assured. Besides, these representative 
 sounds offer so little scope for artistic modification and construction 
 that their introduction into a musical work is obviously inappropriate 
 and produces a ludicrous effect. 
 
 But there is an aspect of the question that seems to be of much more 
 general importance than these. Sounds cannot really be said to represent 
 the animal or object that emits them, any more than a cloud of dust 
 represents a motor-car, or a trail of steam a railway engine. The form 
 of any object represents it or is it in an intimate way that characterises 
 no other effect produced by it. For in all these the form presented 
 in the effects differs entirely from the form of the object itself. In the 
 case of sound this is especially so. The formal aspects of sound (tone, 
 volume, and interval), and the lines of melodies have no relation at all 
 to the shape of the objects that emit them. They cannot then properly 
 represent them. But the pictorial representation of a house or of a 
 man does not merely make us think of the house or the man; it is 
 (pictorially) the house or the man. 
 
 It is in no way the merit of music that it excludes representation. 
 It does so merely because few things produce more than a characteristic 
 noise or timbre, and those features of sounds are the hardest of all to 
 reproduce exactly. In short music cannot be representative. Pictorial 
 art, on the contrary, cannot readily exclude the representative aspects 
 of things without sacrificing much that is of the highest interest and 
 so impoverishing itself greatly. It can, of course, limit its scope to the 
 purely ornamental; but valuable and beautiful as that is, our interest 
 in representative forms is so intense that we constantly long to see the 
 forms of pure ornament vivified by the use of representation. 
 
 There is, therefore, a comparatively limited field for the art of sound 
 to build upon, great and beautiful as are the edifices it raises thereon. 
 Between this field and the emotional life that is expressed in the art 
 there is a direct and continuous transition. The claim of music to be 
 the standard of all art implies that in the pictorial arts all the ranges 
 of experience other than those of colour and form that enter into artistic 
 appreciation are unlawful intruders. Their presence is, as it were, a
 
 xxiv] AESTHETICS AS A PURE SCIENCE 227 
 
 concession to our prevailing habits of mind, not a proper part of the 
 art's action. The probability of this being true is, to say the least, 
 extremely small. 
 
 For it is the peculiarity of the aesthetic attitude that it is always 
 direct; it appreciates an aesthetic object only in so far as it is before 
 the mind, not at all in so far as it is representative. Of course, representa- 
 tion is not excluded; but the material thus brought before the mind 
 is of value only in so far as it is what it is, not in so far as its purely 
 representative function is concerned. The primary basis of music is 
 undeniably the tonal material and all its forms and movements. It is 
 not easy to make these representative without effort; but beautiful 
 music might conceivably please merely because it reminded the listener 
 of some beloved person who once made such music or because it indicated 
 much skill and energy in the player. Many concert-goers are inclined 
 to judge musical performances from the point of view of their acrobatic 
 perfection and virtuosity, if not even from that of the loudness of sound 
 attained. In the same way the mere colours and forms of ornament 
 may suggest many extraneous ideas to the spectator. But neither of 
 these deviations from the artistic attitude make us desire to get rid 
 of the suggestive sensory material from the arts. It is not forbidden 
 that a piece of music shall be difficult to perform, but only that it shall 
 not be performed so that its technical accomplishment stands forth as 
 the chief object of interest to the discredit of the inherent beauty of 
 all its sounds. The patchy bedabbled canvasses that are often seen in 
 exhibitions err in a similar way by making the method of work so 
 obvious that the eye is distracted from the intended beauty. But no 
 one would desire that the method of work should always be so hidden 
 away as to be undetectable, merely because there are many who might 
 rather observe it than the picture's beauty. Why then should we wish 
 to banish the representative functions from visual arts because some 
 folke may take the work as an illustration and not as art? 
 
 It is surely a great mistake to try to limit the basis of construction 
 of any art to a certain range of 'experience,' even if it be to the sensory 
 range. On what grounds shall we fix our limits? Take, for instance, 
 the space of vision that is so important even in the flat arts of sight. 
 A solid sculpture seems to be the only solid type of artistic object. 
 A picture cannot really be solid; its space is only indicated; if it were 
 not so, we should take it for a solid 'panorama,' as we see a room 
 reflected in a mirror. The picture's space is only perspectival and 
 'tonal,' in both forms 'suggested' to the eye, not given. But space is
 
 228 AESTHETICS AS A PURE SCIENCE [CH. 
 
 one of the most powerful interests of pictorial art, atmosphere, 
 distance, roominess, etc. Is it for that reason a foreign element in the 
 art? Certainly not : for the simple reason that the spaces of art are 
 beautiful, as beautiful as are the lines, areas, and forms. If we can look 
 upon the latter without relation to the real distances and shapes of 
 physical things that they may resemble, we can just as readily survey 
 the space of a picture without relation to the cubic yards and miles 
 of physical space to which it may be equivalent. 
 
 And, then, what sort of an object for aesthetic contemplation would 
 a solid sculpture or bronze be that had no resemblance to any known 
 solid object, that, in other words, was devoid of all representative 
 force? If it were not purely ornamental, it would tend to lack the 
 coherence of design. If an architecture is not merely a pretty symmetrical 
 surface, it must be welded upon the design of the building it covers, 
 so that it expresses its functions beautifully, as a beautiful body shows 
 the graces of the human frame. It is a distinctly inferior art that 
 ignores the internal body and merely conceals it within a superficies 
 of beauty. Why should not doors and windows and turrets be moulded 
 upon a rocky hillside instead? But if they belong sensibly only to the 
 cathedral or to the house, we must each time make an imperishable 
 unity in beauty of the inner demands and the outer show. The designs 
 of solid art thus inevitably find their springs in the designs our furniture, 
 our houses, our bodies and our clothes require. Where then is the 
 artistic irrelevance of such things of the world? How can we desire to 
 exclude them from our art and to make its only content the beauty 
 of ornament? It is not given to us to carve the mountains. But when 
 they reveal their design in beautiful forms, awesome and uplifting, are 
 they too not works of art, even though the artist be nature herself? 
 
 The only escape from this conclusion involves the acceptance of an 
 essential irrelevance in all art, namely that what is beautiful is not what 
 is before us in the aesthetic object, but only the fact that that object 
 is the outcome of the artist's desire to express himself. That is out and 
 out heteronomy. The only beautiful way in which a person can express 
 himself is obviously to express himself in himself. Then he and his 
 expression are one in perfect coincidence, and beautiful. But when 
 a man makes a work of art, he makes an object that expresses itself 
 as independently of him thereafter as his grown son ever could. In 
 the ideal creation the artist's personality would be as completely 
 indiscernible as is the hand of God in nature. The works of such a 
 man would really create themselves; they would spring into being in
 
 xxiv] AESTHETICS AS A PURE SCIENCE 229 
 
 their fundamental nucleus of purpose or design, and they would clothe 
 themselves merely by the unfolding and complication of that first 
 germ. We may well believe from many indications that the greatest 
 works of art have thus come into being. The greatest artist in his greatest 
 moments seems not to mould and to form his works but merely to yield 
 himself to the impulses of artistic force. He is not so much a maker as 
 a discoverer of beauty, however much he may have to grope and to 
 search before he finds the true beauty. Its truth has no relation to the 
 length or manner of his search. His sole task is by some means or other 
 to find the true beauty and to recognise it then. 
 
 When the older writers said that art's function was to imitate nature, 
 they may not have been so far from art's secret as we are now usually 
 inclined to think. We know, of course, that the beautiful Venus is not 
 found amongst living women and then merely portrayed. But we must 
 not forget that she has the whole design of woman, that she expresses 
 the ideal of woman as she lives and moves, and that her great value 
 for us lies in that, as well as in her artistic embodiment in stone. In 
 the Venus we do, in a sense, really 'imitate' nature; we take her design 
 and the surface she has given it and in the whole we wrap up and 
 preserve for our souls the elusive perfection. Would it be the same if 
 we made an equally beautiful work of an organic type that has no 
 counterpart or kin in nature? We not only lack the underlying design, 
 but we do not even feel the need for its existence. When the need 
 arises, the design soon follows and grows; and in time, with the fullness 
 of experience and understanding, its own beauty permeates the whole. 
 
 Of course we may want to assert that we thus better nature rather 
 than imitate her. There is no harm in this preference, provided we do 
 not in our conceit imagine that even in our highest conceptions we have 
 created much. We have been given almost all we have, even if we have 
 added our labour to it. It is really absurd nowadays to keep up the 
 foolish belief that nature is never beautiful, but only the compositions 
 of man. On the contrary, we now feel that man is the child of nature, 
 and that in our best moments we do but see more clearly what nature 
 means with us in all her doings. It is not art's task, to be sure, merely 
 to illustrate nature. But there could be no greater task for the visual 
 arts than to discover nature's inner designs and to reveal the full 
 harmony of these with the forms and actions into which she weaves them. 
 
 It is not given to music to reveal the ideals of our real world. But, 
 using a Hegelian inversion, we might well claim that music's task is
 
 230 AESTHETICS AS A PURE SCIENCE [CH. 
 
 to reveal the realities of our ideal world. It has often been urged that 
 music expresses much more intimately and purely the life of man's 
 soul than any other art. How it does this may be inferred with some 
 probability of correctness from the results of our analysis of music. 
 We have learned that music is essentially paraphony, an interweaving 
 of simultaneous melodic rivulets. Rhythm is a most important element 
 in the art, though it has not been the object of our investigation at all. 
 The motional connexions of rhythm are very powerful and combine 
 with those of melody to form a more cogent whole than either could 
 supply alone. The variable features of the paraphony and of the rhythm 
 of music are, therefore, thoroughly motional. They may vary in speed 
 and figure (rhythm in the special sense). Music may speak in one voice 
 or in two or more. These may speak a common language, or they may 
 utter different thoughts. Or, when the articulation of all the voices, 
 or of all but one, is reduced in favour of their common sentiment, an 
 atmosphere of harmony or discord in all its colours and changes may be 
 portrayed. Loudness will indicate its strength, gentleness its peace, 
 height of pitch its brightness or gaiety, depth its sadness or gloom. 
 
 With all these means of variation it is only necessary to bring the 
 motions of music into some sort of correspondence with the character 
 of the acts and energies of man for it to be able to express his soul's 
 life. Fast or slow, vigorous or reposeful, sombre or gay, single or dis- 
 tressed, loquacious or reflective, clear or suspended in doubt, no mood 
 can occur that cannot be depicted in its general character and course. 
 And yet there is withal no representation in the art thus far. 
 
 Nor is there any symbolism or convention. We do not agree, as it 
 were, to look upon the activities of a musical work as symbols of our 
 soul's life. They are merely what they are, the motions of melodies 
 and their tonal conjunctions and changes. We enjoy them in the first 
 place because these things are directly enjoyable by us or beautiful in 
 themselves. But they gain in interest and passion by the fact of their 
 natural resemblance to the 'movements' and activities of our soul as 
 a whole, without our necessarily thereby thinking of our soul at all. 
 
 We touch here on a general problem of aesthetics that has often 
 seemed very mysterious indeed. It is the problem of introjection, the 
 projecting into the external phenomena of the senses of moods and 
 sentiments that are known only in so far as they occur in ourselves. 
 Why does a facade or a trellis look excited or calm? How can the sky 
 look angry, or the sunset full of promise and hope? Are we not forced 
 to believe that only some obscure analogy or association brings these
 
 xxiv] AESTHETICS AS A PURE SCIENCE 231 
 
 notions to our minds in contemplating the object and that we then 
 think the object looks the character that it merely makes us think of? 
 
 We may perhaps draw a better conclusion from our own case of 
 music. Music not only seems restless upon occasion, it really is so. 
 Now it moves at one pace, now at another. And the motional connexions 
 of melody are as much motions in their own way as visual motions are 
 in theirs. They are tonal motions : not spatial, it is true, as visual 
 motions are, but, apart from this difference of cognitive reference, 
 there is no great phenomenal difference. And art is concerned only 
 with the phenomenal; it is not a practical or a scientific issue. 
 
 Again, music not only seems single or involved, it is so, quite as 
 much as a train of thought or of sentiment can be single or involved. 
 Sentiments and tones can both unite into one functional series, giving 
 an effect of arrest or of satisfaction (of tone or of sentiment); or they 
 may interlace in disagreement, producing a confusion that prompts 
 us to seek their (re-)solution. And so on. 
 
 In all these respects we are merely observing and describing in the 
 music what can also be observed and described in the thoughts, senti- 
 ments, and activities, upon which we feel we stake our whole personality. 
 The basis of affinity between the stuff of the art in its changes and our 
 personal mental life is therefore clearly real and indisputable. 
 
 When we feel that a person is angry with us, we do not observe 
 that person's anger, but only his expressions. When we in turn are 
 angry with any one, we not only feel the anger, but we express it and 
 observe whether our expression of it seems to us to coincide with our 
 feeling as we desire it to do. There is an inner agreement between 
 the two that satisfies us. It has often been supposed that only at 
 this moment do we become aware of what anger means. That is 
 surely a mistake. It seems probable that we could equally well learn 
 for the first time what anger is, in the expression given to it by some 
 other person. If it were not so, a savagely angry person might well 
 walk up and finish us off in our innocence. What less is there to note 
 in such a first-felt anger than in the first-expressed rage of our own? 
 Do we, this first time, merely note how our rage expresses itself, being 
 entirely unprepared for the form it takes, and then learning what 
 conduct signifies rage? If our expression turned out to be all smiles 
 and compliments, should we think these were the expressions of rage? 
 I cannot conceive that any mind could be built on so silly and witless 
 foundations. Such a theory hands over all the dignity of coherence 
 to the body and its heredity, making mind a pure farce. Why the body
 
 232 AESTHETICS AS A PURE SCIENCE [CH. xxiv 
 
 should be held to be thoroughly subject to all-pervasive law, while 
 the mind is a mere epi-phenomenon, whose connexions even with the 
 body are completely devoid of inner sense or of continuity with the 
 body, I have not for long been able to appreciate. 
 
 We can therefore in all probability just as well obtain our first 
 experience of anger from the expression (of it) that a person bestows 
 upon us as from the acts of theirs that rouse us to an anger that we 
 then proceed to express. We might, indeed, even encounter some emotion 
 or sentiment for the first time in a work of art, just because that work 
 embodied the sufficient basis of such an emotion. 
 
 From this point of view we can see not only the affinity of music 
 with our soul's life, but also with other arts. The connexion with dancing 
 and all its motions is obviously so close as to be almost a continuity, a 
 complete fusion. Where any other visual display, e.g. that of the stage, 
 can keep properly in touch with the speed of change of music, there 
 can also be an intimate union of the two. But in opera great concessions 
 have often to be made to the stage, so that a charge of incoherence 
 may here be well founded. Music and sculpture or painting do not 
 cohere at all; for the one is an art of succession and of motion, while 
 the others are arts of simultaneity and of motionless form ; and between 
 these all true correlation of detail is lacking.
 
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 (38) MANSFIELD, 0. A. The Student's Harmony. 9th ed. London, 1910. 
 
 (39) MIES, P. Ueber die Tonmalerei. Ztsch. Aesthetik, 1912, 7, 397-450, 578-618. 
 
 (40) MILLER, D. C. The Science of Musical Sounds. New York, 1916. 
 
 (41) MYERS, C. S. The Ethnological Study of Music. Among Anthropological Essays 
 
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 (42) NIECKS, F. Programme Music in the Last Four Centuries. London, 1906. 
 
 (43) GETTING EN, A. v. Harmoniesystem in dualer Entwickelung. Studien zur Theorie 
 
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 (44) OUSELEY, F. A. G. On the Early Italian and Spanish Treatises on Counterpoint 
 
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 2nd ed. 1904. 
 
 (46) - - Art. 'Harmony,' ibid. 1906. 
 
 (47) - - Art. 'Resolution,' ibid. 1908. 
 
 (48) Style in Musical Art. London, 1911. 
 
 (49) PEARSALL, R. L. DE. An Essay on Consecutive Fifths and Octaves in Counterpoint. 
 
 London, 1876. 
 
 (50) POLE, W. The Philosophy of Music. London, 1879. 
 
 (51) POTTER, A. G. Modern Chords explained. London, 1910. 
 
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 1901. 
 
 (53) RAMEAU, J. P. Generation harmonique ou traitd de musique theorique et pratique. 
 
 Paris, 1737. 
 
 (54) Extrait d'une reponse de M. Rameau a M. Euler sur Videntitedes octaves.... 
 
 Paris, 1753. 
 
 (55) ROCKSTRO, W. S. Article 'Hidden Fifths and Octaves,' in Grove's Diet. Music 
 
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 (56) ROUSSEAU, J. J. Dictionaire de la musique, 1764. 
 
 (57) SACCHI, G. Delle quinte successive nel contrappunto e delle regole degli accompagna- 
 
 menti. Milano, 1780. 
 
 (58) SHINN, F. G. A Method of Teaching Harmony based upon Ear Training. Pt. I. 
 
 Diatonic Harmony. Pt. II. Chromatic Harmony. London, 1904-5. 
 
 (59) 2nd Article 'Hidden Fifths and Octaves,' in Grove's Diet. 2nd ed. 1906.
 
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 (60) SHTRLAW, M. The Theory of Harmony. London, 1917. 
 
 (61) STATNER, J. A Treatise on Harmony. London, N.D. 
 
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 1912. 
 
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 INDEX OF AUTHORS 
 
 D'Alembert, J., x, 36 f., 66, 99 f. 
 Aristoteles, quisdam, 186 
 Aristotle, 6, 38, 51, 82 f., 224 
 Aristoxenus, xi, 7, 107, 158 
 
 Bacchiua, 157 
 
 Becker, E., 167 
 
 Boethius, 38 
 
 Bridge, F. and Sawyer, F. J., 179f. 
 
 Browne, W. D., 166 
 
 Cherubini, M. L., 92 
 
 Cotto, J., 82 
 
 Cross, C. R. and Goodwin, H. M., 29 
 
 Cummings, W. H., 86 
 
 Curwen, J., 39 
 
 Day, A., 42, 99, 116ff. 
 Debussy, C. A., 16, 140 
 Descartes, R., 23, 112 
 Dunstable, John of, 86 
 
 Elert, K., 120 
 Euclid, 38 
 Euler, L., 38 
 
 Gardiner, W., 12 
 
 Gaudentius, xi, 15, 108, 157 ff., 213 
 
 Gevaert, F. A., 38, 51, 52, 84 f., 108, 146, 
 
 155, 157, 159 
 and Vollgraff, J. C-, xi, 6f., 52, 82, 
 
 85, 154, 157, 167, 178, 184, 224 
 Gladstone, F. E., 84 ff., 88, 93 ff., 117f., 120 
 Glyn, M. H., 149 
 Guido, 85 
 
 Hauptmann, M., 142 
 Heffernan, J., 24, 33 
 Helmholtz, H. v., vi, 3, 24 ff., 32 f., 69 f., 
 
 87ff. 146f., 162, 176f., 184, 186ff., 
 
 224 
 
 Holden, J., 38, 175 
 Hornbostel, E. v., 153 
 Hucbald, 85 
 
 Hull, A. E., 16, 94, 117, 120, 140, 159, 166 
 Hullah, J., 63 f,, 111 
 
 Jadassohn, S., 125 f., 128 f. 
 
 James, W., 69 
 
 Johannes de Garlandia, 159, 187 
 
 Kaestner, G., 196 
 Kemp, W., 142, 175 f., 186 
 Kitson, C. H., 124, 126, 128 
 Kollmann, A. F. C., 94 
 Krueger, F., 186 f., 195 
 Kiilpe, 0., 175, 186 
 
 Lasus, 7, 38 
 Lee, V., 225 
 Lapps, Th., 26, 38 
 
 Macfarren, G. A., 3, 51, 85 f., 89, 93, 97 L, 
 
 11 6f., 133 f., 136, 166, 175, 192 
 Mach, E., 100 
 Malmberg, C. F., 195 f. 
 Maltzew, C. v., 52 f. 
 Mansfield, 0. A., 117, 135f. 
 Meyer, M., 42, 188 f. 
 Mies, P., 226 
 Miller, D. C., 3, 27 
 Myers, C. S., 34 
 
 Nicomachus, 38, 154 
 Niecks, F., 226 
 
 Oettingen, A. v., 142 
 Ouseley, F. A. G., 83, 99 
 
 Parry, C. H. H., 104, 116ff., 162 f. 
 
 Pater, W., 225 
 
 Pear, T. H., 175 
 
 Pearsall, R. L. de, 95, 117 
 
 Plato, 83 
 
 Plutarch, 51 
 
 Pole, W., 4,84 ff., 89, 163 
 
 Potter, A. G., 172 
 
 Prout, E., viiif., 35, 99, 102 ff., Ill, 116f., 
 
 121 ff., 128 ff., 133ff., 172f., 177 
 
 184 
 
 Ptolemy, 107 
 Pythagoras, 38
 
 INDEX OF AUTHORS 
 
 237 
 
 Rameaa, J. P., ixf., 23, 36 f., 65 &., 79, 99, 
 
 142, 165 
 
 Riemann, H., 142 
 Rockstro, W. S., 122 
 Rousseau, J. J., 39 
 Russell, B., 100 
 
 Sacchi, G., 66 f., 86, 88 f., 90 ff., 118f. 
 
 Schubert, F., 34 f. 
 
 Scott, C., 132, 140 
 
 Scriabin, A., 16, 80 
 
 Serre, J. A., 28 
 
 Shinn, F. G., 95ff., 116ff., 120f., 125ff., 
 
 135 f., 170f. 
 Shirlaw, M., ix, 23 f., 28, 66 f., 142, 149, 
 
 182, 184 
 Sorge, G. A., 28 
 Stainer, J., 117 
 Stanford, C. V., 162 
 
 Stephens, C., 85, 89 
 
 Stratton, G. M., 49 f. 
 
 Stumpf, C., xi, 6, 15, 23 ff., 32, 42, 51, 70, 
 
 83f., 108, 110, 142ff., 153f., 156ff., 
 
 164, 184ff. 
 
 Tartini, G., 23, 28, 142, 224 
 Tchaikovsky, P., 106, 116, 123ff.,$128f., 
 
 135f. 
 
 Thrasyllus, 157 
 Tovey, D. F., 54, 129, 151, 164, 182 
 
 Wagener, A., 84 
 
 Wertheimer, M., 78 
 
 Westerby, H., 142 
 
 Westphal, R., 7 
 
 Wooldridee, H. E., 51, 82 f., 92, 159 
 
 Zarlino, G., 23, 87, 89, 142
 
 INDEX OF SUBJECTS 
 
 Abstraction, 73 f., 77, 83, 87 
 
 Aesthetics as science, vi, 98 f., 131 ff., 221 ff. 
 
 Analysis and synthesis, Iff., 55 ff., 143 f., 
 
 202 f. 
 Art and science, 22, 55, 60, 64, 101, 139 f., 
 
 150, 2141 
 Ascent and descent of pitch, 52, 127, 131, 
 
 184 
 Attributes of sound, 5ff. 
 
 Balance, 8f., 19 f., 57, 72 ff., 79, 200; of 
 
 parts, 97; of distinction, 147 f. 
 Bass most prominent, 51, 92, 102, 130 
 Beats, 24 ff., 261, 31, 56, 1801 
 Beauty objective, 216 ff. 
 Blend, 5,56 ff. 
 Breadth of tone, 7, 131 
 
 Chords, theory of, 148 ff., 163 ff., 209 ff. 
 
 Cochlea, 461, 2031 
 
 Columns of thirds, etc., 791, 148 ff., 163, 
 2091 
 
 Concordance, 141 ff. 
 
 Consecutives, theories about, 8 Iff.; facts 
 about, 101 ff. ; explanation of pro- 
 hibition, 109 ff.; exceptions to pro- 
 hibition, 115 ff.; 1711, 207 
 
 Consonance, grades of, 15 ff., 83, 94, 184 ff., 
 223 
 
 Consonance and dissonance, 15 ff.; Helm- 
 holtz's theory criticised, 24 ff. ; 
 Stumpf's theory criticised, 146 ff.; 
 151 ff., 155; in successive tones, 25, 
 31 ff., 33 ff. 
 
 Convention or habit in music, 84 ff., 89, 93 f., 
 96, 132,2161,230 
 
 Development in music, 151, 62, 92, 96, 113, 
 150, 152 ff., 159, 161 ff., 168, 1961, 
 2121 
 
 Difference tones, 28 ff., 31, 56, 581, 1921, 
 196, 202, 224 
 
 Distinction, loss of, 1471 
 
 Exposed intervals, v. hidden, 1251, 
 
 1361 
 
 Exposure, rhythmical, 134, 1371 
 Expression in music, 229 ff. 
 
 Factors, positive and negative, in beauty, 
 89, 96, 1271, 173 ff. 
 
 Field, the auditory, 48 ff., 531, 2011 
 
 Fifth, the, 19 f., 29, 33 1, 59; a bare interval, 
 83, 96; fifths not equivalent, 77 ff.; 
 consecutive, 80 ff. 
 
 Form and mass in hearing, 43 f . 
 
 Fourth, the, 21, 133 ff., 1771, 1911, 210; 
 as dissonance, 1361 
 
 Fusion, 11, 15ff., 18ff., 33ff., 43, 621, 64, 
 74, 78, 94, 96, 1731, 190; explana- 
 tory value of, 141 ff., 185ff.; grades 
 of, 161, 21 ff., 32, 71, 102 ff., HOff., 
 122 ff., 147 ff., 151 ff., 156, 184 ff., 
 2001; neutral grades of, 108, 1111, 
 1471, 187, 195 
 
 Greek music and theory, 61, 15, 18, 51, 
 81 ff., 92, 1071, 110, 146, 151, 154 ff., 
 167, 178, 185, 206, 213 
 
 Harmonics, v. partials 
 
 Hidden intervals, 122ff. ; v. exposed 
 
 Highest tones, 12, 45 ff. 
 
 Horizontal and perpendicular, 631, 1111 
 
 Imagination, force of, 90 f. 
 
 Individual differences in ear, 61 
 
 Instrumental tones, 1 ff. 
 
 Intensity, 6, 8, 12, 19, 72, 1981 
 
 Interval, 17, 36 ff., 200; in inversions, 71 1, 
 173 ff.; cause of precision of, 1921; 
 nomenclature of, 16; ascending and 
 descending, 52 f.; simultaneous and 
 successive, 53; intrinsic character, 
 95 ff. 
 
 Inversion of chords, 66 ff., 761, 88, 95, 149, 
 1511, 175ff., 203, 209 
 
 Ear, training, 1791; absolute, 189; verdict 
 of, 204 
 
 Longitudinal and transverse, 13 f. 
 Lowest tones, 12, 45 ff.
 
 INDEX OF SUBJECTS 
 
 239 
 
 Major and minor triads, difference between, 
 52, 142, 166 f. 
 
 Materialistic theories, prejudice for, v, 204 
 
 Melody, 205; statistics of, 34 f.; in bass 
 voice, 51 f. ; as surface of music, 54, 
 61; as basis of prohibition of con- 
 secutives, lllff., 126; of hidden or 
 exposed intervals, 128ff. ; of fourth 
 from bass, 134ff. 
 
 Melodic distinction, 148, 156, 160 ff.; flow, 
 172, 178, 182, 206, 208 f.; steps and 
 leaps, 34 f., 124 f., 128 ff., 134 ff. 
 
 Memory, 27 f., 31, 42, 194, 196 
 
 Method, 101 f., 109 f., 122 
 
 Motion, contrary and similar, 82, 170ff., 
 207 f. 
 
 Mystery in music, 140 
 
 Objectivity of beauty, 214 ff. 
 
 Octave, the, 19, 33, 42 f., 59; equivalence 
 
 of, 65 ff., 96, 113f. 
 Order or position, 7, 48 ff., 198 f. 
 Organum, 82 f. 
 Overlapping of tones, 10 ff., 18 ff. 
 
 Paraphony, 155 ff., 187 ff., 206 ff., 230 ff.; 
 as basis of polyphonic music, 160 ff.; 
 factors that modify, 169 ff.; inter- 
 action of, 173 ff. 
 
 Partial tones, viii, Iff., 15 f., 149, 166 f., 
 180 f., 192 ff., 196, 202; relation to 
 fusion, 25 ff., 31; in synthesis and 
 analysis, 56 ff., 61, 67 ff., 76 f ., 87 ff . 
 
 Particles of sensation, 50; of sound, 9ff.. 
 18 ff. 
 
 Parts, the number of, 130 f . 
 
 Pattern, 72 ff., 145, 165, 203, 211 
 
 Pitch, 4, 7ff. ; rising as departure, 52; 
 musical range explained, 45 ff., de- 
 fined, 47; relation to volume, 8ff. ; 
 central in volume, 9ff., 19, 199 
 
 Pleasure in relation to beauty, 217 
 
 Polyphony and harmony, 43 f., 62 ff., Ill, 
 162 ff., 205 
 
 Primary tones in analysis and synthesis, 60 ff . 
 
 Proportion, 36 f., 73 
 
 Pure psychology, 5,20 ff.,37, 54,98 ff.,221 ff. 
 
 Pure tones, 3ff~ 
 
 Quality, 5; amongst the tones of an octave, 
 70 f., 78 
 
 Relationship of chords, 94 ff., 117f., I20f., 
 
 127 ff., 184,212 
 
 Representation in music, etc 225 ff. 
 Resolution, 137, 182 f., 211 
 Rest and motion, musical, 151 ff., 155 
 Rhythm, 91, 137, 178, 230 
 Rules in music, 81, 84 ff., 132, 216 f., v. 
 
 convention 
 
 Sensations, 5 
 
 Seventh, diminished, 106 
 
 Sixth minor and augmented fifth, 180 ff., 210 
 
 Smoothness in tone, 8f., 25, 57 
 
 Sonorous body, the, ix, 66 f., 149 
 
 Statistics, viii, 34 f., 115, 117ff., 139 f. 
 
 Subjectivism, 214ff. 
 
 Surface of tone, 57 f . 
 
 Symmetrical, 8f. 
 
 Synergy, 23 f., 32 f., 70 
 
 Thirds and sixths as consonances, 107 f., 
 
 142, 147 f., 154, 157 ff. 
 Tonality, 53, 90 ff., 164 ff., 183 f., 21 If. 
 Tones and noises, 8 
 
 T07TOS, 7 
 
 Trend of opinion, 104, 115, 123, 128, 
 
 131 f. 
 
 Triad, theory of, 148 ff. 
 Tritone, 103", 105 f. 
 
 Unconscious, the, 59 
 Unison, the, 130, 151 
 
 Vision, compared with hearing, 33, 40 f., 
 43, 48 ff., 73, 85 
 
 Voices, relative prominence of, 51, 92, 102 ff., 
 122 ff., 128 f., 208; effect of number 
 of, 127 f., 169 f., 208; llOf. 
 
 Volume, 6ff. ; inversely proportional to 
 ratio of vibration, 20; graded in in- 
 tensity, 72, 198 f. 
 
 Western music, early, 51, 82 f., 151 
 Zones, 93 f.; of paraphony, etc., 188 ff. 
 
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